University  of  California  •  Berkeley 

The  Theodore  P,  Hill  Collection 

of 
Early  American  Mathematics  Books 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/collegerequiremeOOtibbrich 


COLLEGE  REQUIREMENTS 


ALGEBRA 


A  FINAL  REVIEW 


BY 

GEORGE   PARSONS   TIBBETS,   A.M. 

Instructor  in  Mathematics,  Williston  Seminary 


o5»;o 


BOSTON,   U.S.A. 
PUBLISHED  BY  GINN  &  COMPANY 

1892 


Copyright,  1892, 
By  GEORGE  PARSONS  TIBBETS,  A.M. 


Typography  by  J.  S.  Cushing  &  Co.,  Boston,  U.S.A. 


Presswork  by  Ginn  &  Co.,  Boston,  U.S.A. 


PREFACE. 


The  Williston  students  have  found  these  reviews  so 
serviceable  that  a  more  convenient  form  has  become  nec- 
essary. From  a  wide  collection  of  college  papers  about 
four  hundred  examples,  illustrating  nearly  every  prin- 
ciple in  Algebra,  were  selected  and  carefully  arranged 
by  subjects.  Whenever  a  suitable  one  could  not  be  so 
obtained,  an  original  problem  or  one  from  foreign  texts 
was  inserted.  The  parallel  sections  are  for  the  use  of 
two  divisions  and  for  recitation -room  drill. 

Colleges  will  find  the  work  useful  as  an  initial  review ; 
while  college  candidates  may  be  assured  of  entering  if 
they  perform  all  the  examples  without  aid. 

Suggestions   in    regard   to    the   work   will    be   gladly 

received. 

G.  P.  T. 

Williston  Seminary, 
Easthampton,  Mass.,  Jan.,  1892. 


CONTENTS. 


Section  Page 

Sight  Problems I.  7 

Parentheses  and  Evaluation   .     .     .  II.,  III.  8,  9 

Factoring IV.,  V.  10,11 

H.C.  F.,  L.C.  M.,  Evolution  ....  VI.,  VII.  12,13 

Fractions VIII.,  IX.  14,  15 

Simple  Equations X.,  XI.  16,  17 

Simultaneous  Equations XII.,  XIII.  18,  19 

Theory  of  Exponents XIV.,  XV.  20,  21 

Radicals XVI.,  XVII.  22,23 

Quadratics XVIII.,  XIX.  24,  25 

Simultaneous  Quadratics      ....  XX.,  XXL  26,  27 

Inequalities,  Proportion,  Variation  .  XXII.,  XXIII.  28,  29 

Progressions XXIV.,  XXV.  30,  31 

Binomial  Theorem,  Permutations      .  XXVL,  XXVII.  32,  33 

Undeter.  Coef.,  Limits,  Logarithms  .  XXVIII. ,  XXIX.  34,  35 

Specimen  Paper,  Advanced  Problems.  XXX..  XXXI.  36,  37 

Harvard,  Yale XXXIL,  XXXIII.  38,  39 

Vassar,  Wellesley XXXIV.,  XXXV.  40,  41 

Cornell,  Bryn  Mawr XXXVI.,  XXXVII.  42,  43 

Technology,  Sheffield XXXVIIL,  XXXIX.  44,  45 

Princeton XL.  46 

5 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  7 

SECTION   I. 
^SIGHT    PROBLEMS. 

1.  Of  452  students  who  tried  the  examinations  last  June, 

X  were  men,  the  rest  were  women.  In  all,  z  students 
failed.     How  many  of  the  failures  were  men  ? 

2.  The  sum  of  3  consecutive  numbers  exceeds  the  middle 

number  by  10.     What  are  the  numbers  ? 

3.  In  how  many  weeks  will  x  horses  eat  50  bushels  of  oats, 

if  one  horse  eats  y  bushels  in  a  week  ? 

4.  A  is  20  years  old,  and  B  is  —  2  years  older ;  what  is  the 

age  of  B  ?  \^HaTvard.^ 

6.  Two  men  working  separately  can  do  a  piece  of  work  in 
X  days  and  y  days  respectively ;  find  an  expression  for 
the  time  in  which  both  can  do  it  working  together. 

\_Harvard.'] 

6.  A  pole  100  feet  high,  standing  on  the  side  of  a  hill,  breaks 

off  so  as  to  form  a  right  angle  at  the  break,  with  the 
top  resting  on  the  hill  75  feet  from  the  foot  of  the 
pole;  where  did  the  pole  break ?  \_Cornell.'\ 

7.  Two  steamers  ply  between  the  same  two  ports  a  distance 

of  420  miles.  One  travels  half  a  mile  an  hour  faster 
than  the  other,  and  is  two  hours  less  on  the  journey ; 
find  the  rates.  \_Vassar.^ 


8  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION   11. 
PARENTHESES    AND    EVALUATION. 

1.  ?>x  —  {px-[^x-{y  —  x)])~{—x~^ij). 

2.  a  -  {2h  +  [?>c  -  ?>a  -  {a  +  h)]  +  2a  -  {h  +  3c)). 

3.  ax  +  b{x  +  c)  +  c''-  [{a  -b)x  —  {h  —  c){h  +  c)]. 

4.  ah  —  c(x  —  h)  —  [{x -\-  c)(x  —  c)  —  c(h  ~\c  —  x\)  —  x^\ 

5.  -[\Zx-l^y~^\2x--2{-y  +  2x)-ly\]. 

6.  a-f /^^:r-3a(y-2a;)  +  3a;[4a  +  2(45  +  3)]}. 

8.    Evaluate  a  +  2x—\h  +  y  —  [a  —  x  —  (h  —  2y)]\. 


3/- 

X—  -v  X 


9.    Evaluate  x  —  ( Vo;  +  1  +  2) —  when  x  =  i 

X  —  4: 


10.    Substitute  x  +  S  for  y  in  y^  +  2y^-15y  —  36. 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  9 

SECTION   III. 
PARENTHESES    AND    EVALUATION. 

1.    2a-[bb  +  {?>c-{a  +  [2b~Za  +  ^c])\\ 


2.  3a-13a-[3a-(3a-3a  -3a)-3a]  — 3a|-3a. 

3.  {a  +  h)x  —  (h  —  c)c  —  [{h  —  x)h  —  (5  — c)(a  +  c)]  —  ax. 

4.  3^-4y  +  5[-4.'r-{3y-(2+7a;-2y)-4|]. 


5.  3c'  +  c(2a- [6c-{3a  +  c-4a|]). 

6.  o^f  -f-xy'^  +  a^-  2- j  xy  -  ^'[-  \y^  -  3/(0:3/  -  a;^)}]. 

7.  (a^  -  5=^) c  -  (a  -  5)(a[Z>  +  c]  -  Z>  [a  -  c]). 

8.  Evaluate  -{/a  +  -v/a^  +  Vh  when  a  =  8,  5  =^  64. 


9.    Evaluate  V~a^h'+  ^^l±l^±^  1  ^hen  a^S,b  =  l. 


10.    Substitute  3/  -  3  for  :r  in  :r^  +  2  a;'  ~  15  a;  —  36. 


10  COLLEGE  REQ  UIREMENTS  IN  ALGEBRA. 

SECTION    IV. 
FACTORING. 

1 .  Difference  of  Squares  :  {x^  +  y^  —  z^y  —  4  x^y\ 

2.  Cubes:  x^ —  \. 

3.  Simple  Trinomial :  x"  -  \?>xY  +  362/*. 

4.  Complex  Trinomial :  ^x^  —  bxy  —  ^y^. 
\   6.  Literal  Trinomial :                             x^ -\-{a^-\xy -\-y'^. 

6.  Four  Terms  :  a*  +  a^h''  —  h'^c'  -  c\ 

7.  Five  Terms:  ^a" -\-bah —  ^h' —  ^ac  +  ^hc. 

8.  Six  Terms:  2cf  -  ah  -  ac  -  ^h' -  bhc  -  c\ 

9.  Perfect  Square  :  16:r^+ 16a;' -  4:r^  -  4a;^  +  ^''. 
10.  Imperfect  Square  :  x^  —  7:ry  +  ?/*. 

^11.  Radicals:  rr^  +  l- 

12.  Separation  :  ^-^  —  6^^  +  11  :r  —  6. 

13.  Literal  Exponents  :  x""^  +  ^x"^  +  ^V- 

14.  Parentheses  :  f  1  +  y^  -  2.r-  fl  -  ?/)  +  ^* (1  -  .v)'- 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  H 

SECTION  V. 
FACTORING. 

1.  Difference  of  Squares  :  1  —  {pc^  +  y^)  -f-  ^xy. 

2.  Cubes:  o}'' -'b'\ 

3.  Simple  Trinomial :  x^  —  2x—?>. 

4.  Complex  Trinomial :  Sa'"*  +  Scz  —  3. 
6.    Literal  Trinomial :                       1  —  (m^  +  'n!')x^  +  m^yfx^. 

6.  Four  Terms  :  m^x  +  m^y  —  n^x  —  n^y. 

7.  Five  Terms  :  ao(f  —  ^  ax*  +  2ax^  +  ax'^  —  ax. 

8.  Six  Terms:  0)0"  -  bah +  l^ac  +  h'' -  bhc  + ^c\ 

9.  Perfect  Square  :  67^;'^ -f  49  +  9:^:*  -  70:^:  -  30rl 

10.  Imperfect  Square  :  9a*  -  40a'Z>'  +  16//. 

11.  Radicals:  a  +  b. 

12.  Separation:  a;' +  10:^:'  + 29^+ 20. 

13.  Literal  Exponents  :  x^  —  3/". 

14.  Parentheses:     a{a-~V)x'^  —  {a —  b  ~\)xy —  h(h +  \)y'^. 


12  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION    VI. 
H.C.  F.,    L.  C.  M.,    EVOLUTION. 

Express  both  H.  C.  F.  and  L.  C.  M.  in  factors :  — 

1.    6^'^+ 13a; -5, 

Zx^  +  2x''  +  2x  -  1.  [Princeton.'] 

U'-I2h-S.  [Yale.'] 

3.  ^x'  ~-V^x''  +  ?>x''  +  2x, 

^x'-^x'  +  lbx^-21x-  9.  [Brown.] 

4.  4:x'—9x'  +  ^x-l, 

6^^  —  1  x^  -{- 1.  [Johns  Hopkins.] 

5.  ^x'-bx^-lOx'  +  ^x-lO, 

4iX^  —  4:X^  —  ^x  -\-b.  [Harvard.] 

6.  8:r'  +  27, 

16^'*  + 36a;' +  81, 

6  a;'  +  5  a;  —  6.  [Vassar.] 

7.  Square  Root  of  4a*+ 12a'  + 5a' -  6a  + 1. 

8.  Cube  Root  of  8a6+ 12a^  -  6a*- lla^  +  3a'  + 3a  -1. 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  13 

SECTION   VIL 
H.C.  F.,    LC.  M.,    EVOLUTION. 

Express  both  H.  C.  F.  and  L.  0.  M.  in  factors  :  — 

1.  ^a^-llx"-^, 

^oif" +  11x^  +  ^1.  [Johns  Bopkins.l 

2.  8a;' +  2a; -3, 

12a;^  +  10a;'-4.  [Amherst] 

3.  15 aV  -  20 aV  -  65 a'x  -  30 a^ 

12  bx^  +  20  bx'  -lQbx-16b,  [Brown.] 

4.  a;='-3a;-2, 

a;'  -  2 a;*  -  a;  +  2.  [DartmoiM.] 

5.  o-'^-l, 

a;'' -2a; -3,    - 

6  a;' -a; -20.  [Technology.] 

6.  2a;*  +  a;'- 8a;' -a; +  6, 

4a;*  +  12a;'  -a;' --27  a; -18, 

4^.4  +  4^3  _  17 -^2  „  9^  _^  13  [Harvard.] 

7.  Square  Root  of  8a;'  +  — -  8a;*  + 2a;' - -+ 2. 

8a;®  a;' 

8.  Cube  Root  of  x^  -  6a;^  +  3a.*  +  28a;'  -  9a;'  -  54a;  -  27. 


14  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION  VIII. 
FRACTIONS. 


1.  Reduction  : 

2.  Addition  and  Subtraction  : 

d^  —  be 


4^:^  +  3:1; -10 


.         IP'  +  ca         .         c^  -\-  ah 


(a  -  h){a  -  c)      (5  -f  c)(h  ~  a)      (c  —  a)(c  +  b) 


3.    Multiplication 


X  -\-  y  _  X  —  y  _     4  ?/■ 
_x  —  y      X  ^y     X 


'-M 


x  +  y 
22/ 


4.    Division  : 


^2  _|_  y2  ^  ^2  _  y: 

x^  —  y'^      x^  +  y 


n  by  r^+^^" 

'J      V^  +  y    ^  -y_ 


5.    Complex  : 


12  + 


1  +  - 


2  +  ; 


6.    Evaluation 


If  0;  =  — — -,  find  the  value  of  — —^ — | 


a  +  V 


x  —  2a     x  —  2h 
1  —  a;  V    ^ 


7.    Miscellaneous: 


l-^y\l+x 


x^  +  y^  —  X  -^  y 


\-y  l-f 


COLLEGE  REQUIREMENTS  LN  ALGEBRA.  15 

SECTION   IX. 
FRACTIONS. 


1.    Eeduction  : 


2.    Addition  and  Subtraction  ; 


1      2x^-\-\lx'-^^x~-2^ 
X      1407^-310;^ -31a; -6 


^       -+  .     \     ..+■      1 


x{x  —  a){x  —  b)      a{a  —  x){a  —  b)      b(b  —  x){b  —  a) 

3.  Multiplication:  ^  -  y  ^ca-^  cy  ^b^  +  y\b 

^  a^  +  y'      b''  +  by     h'  +  y'      c 

4.  Division  :             \^-=^  -  1^1  by  [1+^  +  i+^l- 
6.    Complex:  i 


x  + 


1       ^      - 


2:r+l 


6.    Evaluation  :  If  -^—  =  a,    -^  =  b,   -^- 


y  +  z  x  +  z  x-^y 


=      Cy 


find  value  of h 


a      ,      b       ,      c 


7.    Miscellaneous:  __^^ a^ +  b^ 


1  +  a      \  +  b      \-\-c 


a'M 


1  /-^      a -6 


a-\-b      2\        a-\-b 


16  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION   X. 
SIMPLE    EQUATIONS. 
7a;  +  9     /       2x--V^ 


1. 


'-{- 


5.r  +  2      fr^      ?>x  -  1\  _  3:r  +  19      fx  +  \      ^ 


^     83;  +  23      bx  +  2_2x  +  ^      ^ 
20  3a,' +  4  5 


4.   ^^^^-A(y-4)-|(y-6)  +  A. 


6.    12^30; -. 25  (:r- 4) -.3  (5a; +  14)  J  =  47. 
X      a  —  hex       X       ac  ~  4ibx 


6 


2         2bc         6c  Sbc 


7.   _^_3_)._^  =  _£__2a(2-3a). 
2a  4a=^      3a'^  ^  ^ 


8.    At  what  time  between  7  and  8  o'clock  are  tlie  hands  of 
a  watch  (a)  together  ? 
(5)  opposite? 
(c)  at  right  angles  ? 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  17 

SECTION   XL 
SIMPLE    EQUATIONS. 
bx 


1.    2:r      .  -„  ^       , 

4 


2.   3^-^^-4  =  ^^±li. 


3.   ^+10-g(3.-4)+^3"-^X^"-^)  =  x'-g- 
3  5^^  6  15 


■i)-s('-i)+5('-i)^»- 


.      1  /        a\      1  /        a\  ,   1  /        d\ 

2\ 


5.    3.3a;--'^^^~-^^=^la;  +  9.9. 


^     ax  —h      1  —  X  ,  ^ 


7.    S ax  —  2bx  —  ^c  —  \mx  =  ^c  +  ^7nx  —  n  —  bx  +  2ax, 

6.  A  man  bought  a  horse,  and  expected  to  sell  it  at  10  % 
profit ;  but  had  to  sell  it  for  $50  less  than  he  expected, 
and  then  found  he  had  lost  15  %  on  what  it  cost  him. 
What  did  he  pay  for  the  horse  ? 


18 


COLLEGE  REQUIREMENTS  IN  ALGEBRA. 


SECTION   XII. 


SIMULTANEOUS    EQUATIONS. 


ax  -\-hy  =  c  ^ 
a!x  +  h^y  =  c'  ) 


Sx     57J     9 
bx     Sy     4 


1.  Simple  : 

2.  Reciprocals : 

3.  Parentheses : 

4.  Three  Quantities  : 

5.  Fractions: 

6.  Numerical:  x  —  2y  +  Sz  =  2 

2x-Sy+    z  =  l 
Sx-    y  +  2z  =  9 

7.  Problem.     A  firm  has  Java  coffee  at  a  cents  per  pound 

and  Mocha  at  b  cents  per  pound.  How  much  of  each 
is  there  in  a  mixture  of  a  —  b  pounds  if  it  can  be  sold 
at  c  cents  a  pound  without  loss  ? 


(a  +  b)x  —  (a  —  b)y  =■-  ^ab  | 
{a-b)x-{a+b)y  =  0      I 


ax  -^by=l 
cy  +  dz  =m 
ex  -{-  fz  =  n 


711        71  _ 

a 

X      y 

> 

n  _.  m  __ 

b 

X      y 

COLLEGE  REQUIREMENTS  IN  ALGEBRA. 


19 


SECTION  XIII. 


SIMULTANEOUS    EQUATIONS. 


1.    Simple  : 


6.   Numerical : 


x  +  y 


__2(a'  +  b')  ^ 


x-y- 


4:ab 
:i'  -  b' 


2. 

Keciprocals : 

7    _^    4    _ 
Vx     Vy 

'-+  '  - 

■yjx     Vy 

-4 
=  1 

3. 

Parentheses : 

7(:.  +  y)  +  30 
7(^  +  2/) -3(. 

4. 

Three  Quantities : 

y  +  z=a- 
x+z =h  ' 
x  +  y  =  c  ) 

5. 

Fractions : 

a      h 

a     c 

b      c          I 

y)- 


2x+  4:y  +  21  z  =  28 
7x-  3y  =  Wz=  3 
9x-l0y  =  SSz=   4 


7.  Problem.  A  and  B  walk  in  a  circle  whose  circumference 
is  C.  If  they  start  from  the  same  point  and  go  in 
opposite  directions  they  meet  in  4  hours.  If  they  start 
from  opposite  points  and  walk  around  the  circle  in  the 
same  direction  they  meet  in  8  hours.  Find  the  rate 
at  which  each  one  walks. 


20  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION  XIV. 
THEORY   OF  EXPONENTS. 

1 .  Discussion  :  — ;  x^. 

2.  Proof:  (a"*)"  =  a"^ 

3.  Removal  of  neejative  exponents  :  ~  ^    ~  V 

4.  Combination:  ^^¥  x  ^^V^^  ^?¥. 


s)'x(^rKf)' 


5.  Multiplication  :         (o^  —  o^h^  +  a^b^  -  ah  +  ah^  -  b^ 

by  (a*  +  Z)*). 

6.  Division  :  x'^y~^  —  2  +  x'^y'^  by  x^y~^  —  x'^y^, 

7.  Involution:  \^2_^/'v^y  \ 

8.  Evolution  : 

1  +  4y'*  -  2y"^  -  4y-^  +  25y"*  -  24:y'~^  +  16y-l 


9.    Reduction : 


^2_,_4_l_4^-§ 


10.    Problem.     Find  the  number  whose  cube  root  is  one-fifth 
of  its  square  root. 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  21 

SECTION   XV. 
THEORY   OF    EXPONENTS. 

1.  Discussion:  |^^ ^^^^n.-n       J"' 

2.  Proof:  a^=l,    «-"  =  -• 

a** 

3.  Removal  of  negative  exponents  :  — ^       .,    ■ 

c~*  —  a~^ 

4.  Combination  :  a^y^  ^  (   i )  "^    i ' 

5.  Multiplication  : 

(a^  -  a*  +  1  -  a"*  +  a"^)  by  (a*  +  1  +  a"*). 

6.  Division:  ^'''~'^' "^  f^T^'* 


7.    Involution  : 


•^x-y^^-(y-x)-|-y 


8.   Evolution:  a? +  2x^ -Zx"  -  ^x^  +  4:x. 


9.    Keduction : 


a;^  — 8a:^y 


10.    Equation  :  V^  =  2V2  ;  find  ^. 


22  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION   XVI. 
RADICALS. 

1.    Reduction  : 


x+\     jx—  1 

x-i^x  +  i 


2.    Addition  and  Subtraction  : 


V9  ab'  -  ^/a'b  +  -J a^h  -  6  a^h'  +  9  ah\ 

3.  Arrangement :  V2^    VS,    "V^o-l-. 

4.  Multiplication:  [a:-i(l  -  V^)][a;  -  i(l  +  V^)]. 

5.  Division  :  ^ .,  __    '    bv  ^- — 

6.  Involution  and  Evolution  :  [2V3  +  3 V2  +  V6]l 


7.    Rationalization  :  — z==- 


8.    Imaginaries  :        V—  9:r*  +  V—  IB,-?;*  —  V—  {x  —  Vf. 


9.    Binomial  Surds  :  V^l  +  12V5. 

10.    Radical  Equation  :  V^  +  2  —  V:r  —  2  =  2a;. 


COLLEGE  REQUIHEMENTS  IN  ALGEBRA.  23 

SECTION  XVII. 
RADICALS. 


1.  Reduction:  V(a^  -  l)(a  -  1)1 

2.  Addition  and  Subtraction  : 

2  Vl25  -  ■^/^  +  ^'''81  -  (-  512)*  +  ^192. 

3.  Arrangement :  V3,    V6,    VlO. 

4.  M.l.ip,io.tio.:   (.-1--|1)(«-1J-|1)(.  +  J-)^ 

5.  Division:  V2l/l-VV2\/i. 


6.    Involution  and  Evolution  : 


7.    Rationalization  : 


'x^/yV 


I  -^  xy) 
V3         2~  V^^ 


2  -  V3      2  +  V-  2 

8.  Imaginaries:  [2 V— 3  —  5 V— 2]l 

9.  Binomial  Surds  :  Vl4  +  4V6. 
10.    Radical  Equations  :                  ^x  +  5  +  V^  -  8  =  V3. 


24  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION   XVIII. 
QUADRATICS. 

1.  Simple:       "  .  12:r'^  +  a;- 1  =  0. 

2.  Literal:  a{x' -x)  +  b{x'' +  x)        ^^ 


a-\-b 


3/ Fractional:  (4a^- 5-)(.^  +  D  ^  2.. 

4.    Parentheses  :  {x-2){x  +  3)(n'2  ^  3 ^  _  4)  ^  0. 


5.  Rationalization  :  ^         =  x  -{-  ■\/x^  —  8. 

^  -  Vo;^  ~  8 

6.  Fractional  Exponents  :  .     x^  —  x^  =  256. 

7 .  Quadratic  Form  :  {ax  —  Hf  —  ^a {ax  —  h)  =  ^ a^ 
IQx 


8.  Radicals:  ^^"^       -VlO^  +  S>=. 

Vl0a;-9  Vl0a;-9 

9.  Cubic  Equation  :  x^  —  x^  —  x-\-l  =  0. 

10.  Formation  : 

Find  equation  with  roots    {a  +  h  —  c)  and  {a  —  h  +  c). 

11.  Problem.    Tristram  is  ten  years  younger  than  Launcelot ; 

and  the  product  of  the  ages  they  attained  in  1890  is 
96.     Find  the  ages  they  attain  in  1908.        \_IIarvard.'] 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  25 

SECTION  XIX. 
QUADRATICS. 

1.  Simple:  91a;'' -  2a;- 45  =  0. 

2.  Literal:  ^  .       "= 71 

a—  2o         a  —  Zo       X 


3.    Fractional : 


.r  +  1      2_a;  +  2 
c         ex     ax  —  bx 


4.  Parentheses  :  {x  -  l)(x  -  2){x'  -  6 a;  +  9)  -  0. 

5.  Binomial  Surds  :  (7  -  4  V3)  o;^  +  (2  -  VS)  a;  -  2. 

6.  Fractional  Exponents  :  4  -y/x  +  Vo;  ^  21. 

7.  Quadratic  Form  :  x^  --2x  +  ^y/x^  ~  2a;  +  5  =  11. 

8.  Radicals:  Vo;  +  <^  +  Vo;  +  Vo;  —  a  =  0. 

9.  Bi-quadratic  :  a;*  —  2  o;^  +  a;  —  2  =  0. 

10.  Formation  : 

Find  equation  with  the  roots      ^^"^  /    and  {h  —  a). 

a  —  0 

11.  Problem.     A  cask  P  is  filled  with  50  gallons  of  water, 

and  a  cask  Q  with  40  gallons  of  brandy,  x  gallons 
are  drawn  from  each  cask,  mixed  and  replaced,  and 
the  same  operation  is  repeated.  After  the  second 
replacement  there  are  8|-  gallons  of  brandy  in  P, 
Find  X, 


26  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 


SECTION  XX 

SIMULTANEOUS   QUADRATICS. 

1. 

Substitute  : 

2x  +  ?>y     =12| 
^x''-2xy=lb) 

2. 

Obtain  xy  : 

x-y    ==      4| 
x'  +  y''^  106  j 

3. 

Let  y  =  vx\ 

x''  +  l^xy  =ll\ 
bxy  —  Zy''^    2) 

4. 

Divide  : 

x^  +  :ry  +  y^  -  28  1 

6. 

Ketain  Fractions : 

1-1   =   0^ 
X     y 

x'     y'           J 

6. 

Yale : 

1^:?1::3:7- 

^    y          ^ 

7. 

Wellesley : 

x"^  +  xy=lb'\ 
xy-f^    2) 

8. 

Michigan  University 

x  +  y  —  ^xy  =      7  1 
x'  +  y'  +  xy    -=133  3 

9. 

Columbia ;  Mines  : 

x'  +  y  +  Sx=7S-2xy) 

y'  +  x 

=  U-Sy    3 

COLLEGE  REQUIREMENTS  IN  ALGEBRA.  27 

SECTION   XXI, 
SIMULTANEOUS   QUADRATICS. 

'  =  17/ 

! 

3.    Let  y  =  vx  :  x^  -j-Sxy  =  —  I 


1.  Substitute  :  ;r  —  83/  =  1 

x^  —  2xy-i-9y' 

2.  Obtain  a:y :  ^  —  y    —10 

x'  +  y'  =  58 


4.    Divide  :  x^  —  xy  -i-y^    =9 

x*  +  xy  +  y'  =  2iS. 


=  12  J 

:} 


5.  Retain  Fraction  :    -  +  -   =7 

X     y 

1-1=25 

x'      y' 

6.  Vassar  :  x^-j-y^  =  56 

7.  Princeton  :  :r^  +  :ry  +  ?/^  =  52  ^ 

xy  —  x^ 


8.    Univ.  Penn.  :  x'^  —  xy  ^  y"^    =  1      | 


9.    Harvard  : 


rr'  +  xY  +  y* 

1 /       y        _  ^  +  6a\  ^  Q 


a:  +  y      \a  (^  —  y)      x^  —  y^j 

y :  (1  x  -2y)  =  {b  -  a) '.  {2a~9h) 


28  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION   XXII. 
INEQUALITIES,    PROPORTION.    VARIATION. 

1.  Prove  that  the  square  of  half  the  sum  of  any  two  quan- 

tities <  half  the  sum  of  their  squares. 

2.  Prove  a'>a  +  -—l  if  a>l. 

a 

3.  What  two  numbers  whose  difference  is  c?are  to  each  other 

as  a  is  to  5  ? 

4.  If  rr  —  3/  is  a  mean  proportional  between  y  and  (y-{-z  —  2x), 

show  that  a:  is  a  mean  proportional  between  y  and  2;. 

5.  Prove  that  if 

2x:y::a:b,    \a~  x  :^a-{- x  \ -.h  —  y.h -{-y. 

6.  Prove  that  a  proportion  taken  by  inversion  is  a  true  pro- 

portion. 

7.  A  varies  jointly  as  B  and  C\  and  A  —  ^  when  -5=3, 

(7-2.     Find  ^  when  ^-5,  (7-7. 

8.  A  varies  as  the  square  of  B,  and  inversely  as  the  square 

of  (7,  and  ^-4  when  ^=--1,  C=2.     What  is  the 

value  of  -^^  +  ^'  when  B  =  2,  (7=2? 
B^  -  A^ 

9.  The  volume  of  a  sphere  varies  as  the  cube  of  the  radius. 

If  3  spheres  with  radii  9,  12,  and  15  inches  are  melted 
into  a  single  sphere,  find  its  radius. 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  29 

SECTION  XXIII. 
INEQUALITIES,   VARIATION,    PROPORTION. 


1.    Prove  that  ( ^^  ,]  <  ah. 

V2  +  V 


2.  Find  the  limit  of  a;  in 

(3a;  +  2)(a:-  3)  >  (;r  +  4)(3a:-  1)  -  3. 

3.  What  number  added  to  2,  20,  9,  34,  will  make  the  results 

proportional  ? 

4.  Find  two  numbers  such  that  their  sum,  difference,  and 

the  sum  of  their  squares  are  to  each  other  as  4,  1,  17. 

5.  If  a:h  :  :  c:  d     show  that 

a:b::  VSa'  +  bc' :  VBb'  +  bd\ 

6.  Prove. that  a  proportion  taken  by  division  is  a  true  pro- 

portion. 

7.  The  area  of  a  circle  varies  as  the  square  of  its  radius  and 

the  area  of  a  circle  is  154  sq.  ft.  when  the  radius  is 
7  ft.  Find  the  area  of  the  circle  whose  radius  is  10  ft. 
6  in. 

8.  The  offing  at  sea  varies  as  the  square  root  of  the  height 

of  the  eye  above  sea-level,  and  the  distance  is  3  miles 
when  the  height  is  6  ft.  Find  the  distance  when  the 
height  is  50  yds. 


30  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION   XXIV. 
PROGRESSIONS. 

1.  Given  a,  d,  and  n  in  an  arithmetical  progression,  to  find  I. 

2.  Given  a,  I,  and  r  in  a  geometrical  progression,  to  find  the 

sum  s. 

[Note.  —  The  above  problems  are  given  frequently.] 

3.  The  first  and  ninth  terms  of  an  arithmetical  progression 

are  5  and  22.     Find  the  sum  of  21  terms. 

4.  Find  the  nth  term  of  the  series  2,  2-|-,  2|-. 

5.  Find  d  and  I  when  a  =  3,  n=^lb,  s  =  ~  165. 

6.  Insert  3  arithmetical  means  between  —  9  and  18. 

7.  Find  the  twelfth  term  of  V2,  -  2,  2V2,  -  4,  etc. 


8.    Sum  the  infinite  series  q  +  ^  "I"  03  +  ^^^• 

000 


9.  The  sum  of  3  numbers  in  arithmetical  progression  is  12, 
and  the  sum  of  their  squares  is  50.  Find  the  num- 
bers. 

10.  If  a  clock  is  constructed  so  as  to  strike  up  to  24,  how 
many  strokes  will  it  make  in  the  revolution  of  the 
index? 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  31 

SECTION   XXV. 
PROGRESSIONS. 

1.  Given  a,  I,  and  n,  in  an  arithmetical  progression,  to  find  5. 

2.  Given  a,  n,  and  r,  in  a  geometrical  progression,  to  find  I. 
[Note.  —  The  above  problems  are  given  frequently.] 

3.  In  an  arithmetical  progression,  s  =  —  ■^,  n  =  20,  a  =  \. 

Find  d. 

4.  Find  the  (2n)th  term  of  1,  3,  5,  7,  etc. 

5.  Find  a  and  n  when  Z  =  — 47,  d  =  —  l,  s  =  — 1118. 

6.  Insert  3  geometrical  means  between  |  and  128. 

7.  Find  the  seventh  term  of  ~  -J.  ^,   -  f ,  etc. 
.  8.    Sum  the  infinite  series  \  +  -^  +  yts 

9.  A  traveller  has  a  journey  of  132  miles.  He  goes  27 
miles  the  first  day,  24  the  second,  and  so  on,  travelling 
3  miles  less  each  day.  In  how  many  days  will  he 
complete  his  journey? 

10.    Find  the  sum  of  all  the  numbers  which  are  less  than  500 
and  are  divisible  by  11  without  a  remainder.       [Yale.] 


32  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION   XXVI. 

BINOMIAL     THEOREM.       PERMUTATIONS    AND 
COMBINATIONS. 

1.  Expand  by  the  binomial  theorem  (^a^  —  f  ^Z- 

2.  Expand  to  4  terms  —  [Yale.] 

Vl  +  x'^ 

3.  Expand  (2a-Sb)-\ 

4.  Obtain  4  terms  la^^ )  • 

5.  Find  the  fifth  term  of  (x~''  -  2y^y\ 

(5  /—         2    \^^ 
V  a^  -  -—  J  [ Harvard.] 

7.  Find  the  term  independent  ofa;in  (Sx )• 

8.  Expand  (1  +  2x'y  to  4  terms. 

9.  Expand  (a'  +  l  +  a-y. 

10.  How  many  different  amounts  can  be  made  up  from  5 

different  coins  ? 

11.  In  how  many  ways  can  7  children  form  a  ring? 

12.  I  have  5  single  volumes  and  a  set  of  3  volumes.     In  how 

many  ways  can  I  arrange  these  8  books  on  a  shelf, 
keeping  the  set  together  and  in  order? 


COLLEGE,  REQUIREMENTS  IN  ALGEBRA.  33 

SECTION   XXVII. 

BINOMIAL    THEOREM.       PERMUTATIONS     AND 
COMBINATIONS. 

1.  Expand  by  the  binomial  theorem  (  a )  • 

V       ay 

2.  Expand  to  4  terms  (a  +  x)'^ . 

3.  Expand  ( VS  —  3  Va)^.  {^Technology.'] 

4.  Obtain  the  first  3  and  last  3  terms  of  {x  —  yf^. 
■  5.  Find  the  fourth  term  of  (2a;  -  ?>y)-\ 

6.  Fourth  term  of  I -^-;i=  —  -a3^"M  •  {Harvard,'] 

7.  Find  the  terms  without  radicals  [  2Va  ~  -1/-  )  • 

8.  Expand  to  5  terms  (1  +  a)^. 

9.  Expand  (e*  —  e"*/. 

10.  How  many  different  signals  can  be  made  with  12  different 

flags  by  hoisting  4  at  a  time  above  each  other  ? 

11.  How  many  combinations  can  be  made  from  the  word 

Payson  taken  3  letters  at  a  time  ? 

12.  At  a  whist  party  there  are  6  ladies  and  6  gentlemen. 

The  host  is  to  play  with  the  most  honored  guest,  and 
the  hostess  with  the  poorest  player.  In  how  many 
ways  can  the  players  be  arranged  if  each  man  has  a 
lady  partner  ? 


34  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION  XXVIII. 

UNDETERMINED    COEFFICIENTS,     LIMITS, 
LOGARITHMS. 


2  +  a;      . 

1.    Expand ' into  a  series  of  four  terms  by  unde- 

1  +  x  —  x^  "^ 

termined  coefficients. 


2.    Develop  — ^t —  into  a  series. 


X 

3.    Separate  — — —  into  partial  fractions. 

\X   —  JL  K  ^  —  Ztj 


4.    Separate into  partial  fractions. 

^  {x-l)\x-\-\f  ^ 


5.    Find  the  limit,  when  x  increases  without  limit,  of 

{x  +  V){x''  -  3) 
x'-Zx 


6.    Prove  log^m  =  _2E«^. 


^    g.      yn    2.372  X  7232  x  .003722 
inipiy     ^_  22.37)(72230000) 

2 

8.    Find  the  value  of  x  in  the  equation  5*  =  30. 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  35 

SECTION   XXIX. 

UNDETERMINED     COEFFICIENTS,     LIMITS, 
LOGARITHMS. 

1.    Expand  into  a  series  by  undetermined  coefficients 

1  +  23; 
\—x  —  x^ 


3    I    /J. 
2.    Develop  to  four  terms  - — — ^^ 

A  —  X  —  X 


3.    Separate  — ^  into  partial  fractions. 


X  —  a 


4.    Separate  ^        — — -  into  partial  fractions. 

^  {x-2)\l-2x)  ^ 


5.    Evaluate  ^— — ^^^^'  +  ^^  for  a;  -  00. 
x'-?> 


6.  Prove  that  logja  X  log„6  =  1. 

7.  Simplify  a/126 VT08^a/1008^T62. 

8.  Given  the  amount  of  a  given  principal  for  a  given  num- 

ber of  years,  to  find  the  rate  per  cent. 


36  COLLEGE  REQUIREMENTS  IN  ALGEBRA, 

SECTION   XXX. 
SPECIMEN    PAPER. 

1.  Evaluation:  The  score  of  the  Amherst-Technology  game 

on  Saturday  was 

V  A 
Find  numerical  value  if  e  ~f  =  g  =  h  =  4:. 

2.  Expression  :  A  man's  monthly  salary  is  ^x.     His  weekly 

expenses  are  ^w,  besides  annual  tax  of  ^t,  and  semi- 
annual insurance  of  $5.  How  much  does  he  save 
yearly  ? 


3.  Parentheses:       c-[2a-b-(Sa-2b-Aa-Sb)].    [Sheffield.] 

4.  Multiplication:  (!«"*  + i^"*"' +  i)(i  a  -  i). 

5.  Division:  (a' -2b' -Qc'  +  ab  -  ac +  1bc) 

by    (a-b  +  2c).  [Wor.  Tech.] 

6.  Formulas  :  Square  [(b  —  2)x  +  (1  —  b)]. 

7.  Inspection:  Divide  I a^ )  by  la )      [3fi7ies.] 

8.  Factoring:  x"" -y' —  z" +  2yz.  [Yale.] 

r.     r.    ^r  (12x'-29x  +  U, 

9.  G.  C.  Measure :    -j         ^  [Harvara.] 

V.  lo  ^    "j—     oX —  i.U. 

10.    Miscellaneous  :   H.  C.  F.  by  factoring  or  division 


■x'-(a'  +  b')x'  +  a'b\ 

.x'-(a+  by  x'  +  2ab(a  +  b)x  -  a'b\ 


French 
Collection, 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  37 

SECTION   XXXI. 
ADVANCED    PROBLEMS. 

1.  A  certain  librarian  spends  every  year  a  fixed   sum  for 

books.  In  1886,  the  cost  of  his  purchases  averaged 
two  dollars  per  volume  ;  in  1887,  he  bought  300  more 
volumes  than  in  1886  ;  and  in  1888,  300  more  volumes 
than  in  1887.  The  average  cost  per  volume  was  thirty 
cents  lower  in  1888  than  in  1887.  Find  the  number 
of  volumes  bought  each  year,  and  the  fixed  price  paid 
for  them.     (Obtain  two  solutions.)  [^Harvard.^ 

2.  A  and  B  start  at  the  same  time  from  two  towns  and 

travel  towards  each  other.  When  they  meet  B  has 
travelled  a  miles  more  than  A ;  it  will  take  A  h  days 
longer  to  reach  the  town  B  left,  and  B  c  days  longer 
to  reach  the  town  A  left.  Find  the  distance  between 
the  towns.  [Harvard.^ 

3.  Three  students  A,  B,  and  C,  agree  to  work  out  a  series 

of  difficult  problems  in  preparation  for  an  examination  ; 
and  each  student  determines  to  solve  a  fixed  number 
every  day.  A  solves  9  problems  per  day,  and  finishes 
the  series  4  days  before  B ;  B  solves  2  more  problems 
per  day  than  0,  and  finishes  the  series  6  days  before 
G.  Find  the  number  of  problems  and  the  number  of 
days  given  to  them  by  each  student.  [Harvard.'] 


38  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION   XXXII. 
HARVARD. 


[Write  legibly  and  without  crowding ;  give  the  work  clearly,  and 
find  all  possible  answers.  The  shortest  methods  of  work  are  preferred. 
Abridged  processes  of  work  may  be  used,  but  should  be  distinctly  indi- 
cated.! 


1.    Solve  the  equation 

2  b 


\fjL-.<2)      ^ 


b\2a 


4:a 


U_b      11 
_  X       a     X      b_ 


=  0. 


2.  Two  tanks  A  and  B  are  discharging  water;   A  at  the 

rate  of  x  barrels  per  hour,  and  B  at  the  rate  of 
(x  +  100)  barrels  per  hour.  At  (1  +  3/)  hours  after 
noon,  A  contains  470  barrels  less  than  at  noon ;  and 
at  a  time  {1—y)  hours  after  noon,  B  contains  400 
barrels  less  than  at  noon.  Find  the  rate  at  which 
each  tank  is  discharging  water,  and  the  times  (1  +  y) 
and  (1  —  y)  hours  after  noon.  Obtain  two  sets  of 
answers,  and  interpret  negative  results. 

3.  Find  the  Greatest  Common  Measure  and  the  Least  Com- 

mon Multiple  of  12  :r' -  29^+ 14  and  18  a;' +  3  a;- 10. 

/  Jy  \  81 

4.  Find  the  fourth  term  of  (  Va )  • 

\  Gay 

6.    Find  a  mean  proportional  between 

4a;^  — 3a;— 1    and    x  —  \. 


COLLEGE  REQULREMENTS  IN  ALGEBRA.  39 

SECTION   XXXIII. 
YALE. 

Examination  for  Admission. 

2-^3 3 

1.  Express  in  an  equivalent  fraction  having 

^         1  +  V3-V5 

a  rational  denominator. 

2.  Divide  81  rt;- 16  by  3^7^  +  2. 

3.  Find  the  cube  of  1 -• 

l  +  x^ 

4.  Find  the  square  root  of  27  —  12  V5. 

6.    Solve  the  equation  \  —  *1  x"^  =  2ax  —  h 3(f , 

r  XT/  =  a, 
6.    Solve  the  equations   < 

^  la;''  +  y^  =  h. 


4         . 
7.    Expand — —  into  a  series,  giving  five  terms. 

yCL  —  Zi  Xj 


40  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION   XXXIV. 
VASSAR. 

1.  Prove  that  the  product  of  two  quantities  is  equal  to  the 

product  of  their  G.  0.  D.  and  L.  0.  M. 

Find  G.  C.  D.  and  L.  C.  M.  of 

2.  Simphfy  \ — —t\—^^\ 7\ r 

^    ^     ^a-bMa-b       ^a  +  b\a  +  b 

3.  Solve  the  equations  :  (1)  ^^— ^ —     ~  ^  —  b, 

a  +  -Va^  —  x^ 

(2)  x'  +  f  =  56, 
X  +y  ^2. 

4.  Define  root  of  an  equation.     How  many  roots  has  a  sim- 

ple equation?     Form  the  equation  whose  roots  are  a 
and  —  b. 


5.  Define  a"*,  m  being  a  positive  integer,  and  prove  (a"*)"=a'"". 
Multiply  together  -Vabc,   aH'^c^,    a~H^c~'^. 

Write  down  the  values  of  3*;  5^  0^  16"^;  2-";  4*. 

6.  There   are   three   numbers   in    arithmetical    progression 

whose  sum  is  15.  If  1,  3,  9  are  added  to  them  respec- 
tively, they  are  then  in  geometrical  progression.  Find 
the  numbers. 

7.  Explain  the  meaning  of  ''mean  proportional,''  ''fourth 

proportional.'' 
If  a\b  '.'.  c\d\  prove  that  each  of  their  ratios  is  equal  to 
(a  +  mc)  :  (b  +  md). 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  41 

SECTION  XXXV. 
WELLESLEY. 

1 .    Factor  a;  —  3  a^fx  —  28  d^ ;  m*  +  mW  +  7i* 
and   ah'^c-^  +  a'h-'^c'  -  2. 


2.  simpHfy  r-ji^T 


y        

3.  Add     2Vl25-v'||+ a/si  "(-512)*+ -^192 

and  [7-^9  +  2VT20  +  3(-8/p  +  3 Vi+  7(f )*](V=T;)^ 

4.  Define  simultaneous  equations. 

5.  Sum  the  series  1,  \,  —\ to  9  terms,  and  derive  the 

formula  for  the  sum  of  a  series. 

6.  Solve  for  x  in  the  following  equations. 

1. ?=  +  -  2 


:r+V2-a;2      x--\/2-x' 
2.    3^"-{/^  +  2-^-16. 

7.  Find  a;  and  y  from  the  equations 

(^-y)':^'-2/'  =  3:7| 

xy  =  ^       ) 

using  the  principles  of  proportion  as  far  as  possible. 
Prove  the  most  important  of  the  principles  of  proportion 
used. 

8.  A  distributes  $180  in  equal  sums  among  a  certain  num- 

ber of  people.  B  distributes  the  same  sum,  but  gives 
to  each  person  $6  more  than  A,  and  gives  to  40  per- 
sons less  than  A  does.  How  much  does  A  give  to 
each  person  ? 


42  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION   XXXVI. 
CORNELL 

1.  Define  the  H.  G.  F.  and  L.  G.  M.  of  two  polynomials,  and 

explain  fully  the  theory  of  the  method  of  finding  them. 

2.  Determine  the  common  factors  of 

l^x^-^x'-Zx^^  and  l^x^ +I2x' -~^x -^, 

and  hence  find  3  values  of  x,  which  when  substituted 

in  the  expression 

[{\2x^  -Sx'-Zx  +  2)]  +  [(IQx^  +  12;r^  -  4a;  -  3)] 

will  give  zero  for  the  result. 

3.  If  the  roots  of  the   quadratic   x'^~px  +  q  =  0   are  two 

consecutive  integers,  prove  p^  —  ^q  —  \  =  0. 

4.  Extract  the  square  root  of  47  —  12Vl5,    and  find  the 

value,  when  x  ^=  V3,  of ~ — 

{x-iy      {x-\-lf 

5.  For  what  values  of  m  will  the  equation 

o;"^  —  2  (1  +  3  7?zJ  :i-  +  7  (3  +  2  m)  =  0  have  equal  roots. 

6.  Solve  the  equations  : 

(a) 


x~l      5V^'-1      3y      10(p;-l) 
^  ^   {2bx-2a7/  =  Sb'-a\       ^^^  {  x' +  2i/' -  xt/ =  8. 

8.  A  line  6  in.  long  is  produced.  Find  the  length  of  pro- 
duced part  so  that  the  rectangle  contained  by  half 
the  line  and  the  line  made  up  of  the  half  and  the  part 
produced  may  be  equal  to  the  square  on  the  produced 
part. 


c,     COLLEGE  REQUIREMENTS  IN  ALGEBRA.  43 

SECTION   XXXVII. 

BRYN    MAWR. 

[Students  are  expected  to  answer  questions  in  each  of  the  divisions  B 
and  C  of  the  paper.     Division  A  is  in  Arithmetic] 
B. 
^    5.    Prove  the  identity  : 

(a^  +  h'  +  c'Xd^  +  e'  +f)  -  {ah  +  be  +  cff  =  (hf~  cef 
+  {cd  -  ajy  +  {ae  -  bd)\ 
-  6.    Solve  the  equations  for  all  values  of  x  : 

^  c  ax  —  cy  -\-  cz  —  a  c  ?>b  x'^  -{- ^9  xy  =  x 

(1)  \hx  +  cy  -=b  also  (2)  \  x'  =  4:f 

[ex  —  cz  =  c 


^  7.    Prove  that^  ==^ — —  is  a  root  of 

2a 

ax^  +  bx  +  c  =  0. 

Show  that  if  B  is  the  other  root,  AB  must  =  — 

a 
c. 

^   8.    Explain  as  clearly  as  you  can  why  a~^,  a°,  a"'*,  are  said 

to  be  equal  to  -,  1,  — • 
^        a^ 
"^   9.\'  A  varies  as  the  square  of  B  and  inversely  as  the  square 
of  C.     Also  A  =  ^  when  B=l,   C=  2. 

^^    What  is  the  value  of  ^±-^  when  ^  =  2,   C^-  2  ? 
B'  -  A' 

10.    What  is  the  Z;th   term  in   the   arithmetical   progression 
whose  first  term  is  a  and  nth  term  is  Z? 

Sum  the  series  ~-\ f-  ^+ to  infinity. 

,  How  many  terms  of  the  series  1  +  3  +  5  +  7+  must  be 
taken  in  order  that  the  sum  may  be  (r  —  1)^  where  r 
is  a  positive  integer  ? 


44  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

SECTION   XXXVIII. 
TECHNOLOGY   (BOSTON). 
1.    Reduce  to  their  lowest  terms 


an 


^  a^  +  2a(a  +  l)  +  4 


2.    Show  that 


:v^^'i^M)■-<v^^l)• 


3.    Show  that 


14- V-3V,   /-I- V-~-3^^ 


2  J      \  2 

4.  Solve  the  simultaneous  equations 

:r(y+l)-8;  3/(a;  +  2)-12. 

5.  Find  the  square  root  of 

^6  _  4^5  _|_  20a;2  +  16:r  +  16. 

6.  Solve  the  equation 

7.  Form  the  quadratic  equation  whose  roots  are  -|,  f ,  and  f . 

8.  The  sum  of  three  numbers  in  arithmetical  progression  is 

18.  If  the  first  be  increased  by  1,  and  the  third  by  2, 
they  will  be  in  geometrical  progression.  What  are  the 
numbers  ? 


COLLEGE  REQUIREMENTS  IN  ALGEBRA.  45 

SECTION   XXXIX. 
SHEFFIELD    SCIENTIFIC    SCHOOL. 

Entrance  Examination. 

ALGEBRA    FROM    QUADRATICS. 

[Note.  —  State  at  the  head  of  your  paper  what  text-book  you  have 
studied  on  the  subject  and  to  what  extent.] 

1.  Solve  the  equation  ax^  -{-hx  -\-  c^=0,  and  point  out  what 

relation  must    exist   between  the  coefficients  in  order 
tliat  the  roots  may  be  equaL 

2.  Determine  by  inspection  the  roots  of  the  equation 

3.  Solve  the  equation 

x'-x  +  b^2x^-bx  +  e>  =  ^^'^^^' 

A 

4.  There  are  20  things  of  one  kind  and  10  of  another;  how 

many  different  sets  can  be  made,  each  containing  3  of 
the  first  kind  and  2  of  the  second  ? 

5.  Insert  3  arithmetical  means  between  4  and  20. 

\  —  X        ' 

6.  Expand  :; into  a  series  by  the  method  of  unde- 

termined  coefficients. 

.  (x  -X-  X)(x^ 3") 

7.  Find  the  limit   of   ^^ — — — '-  when  x  is  increased 

a:*  —  3  a; 

without  limit. 


46  COLLEGE  REQULREMENTS  IN  ALGEBRA. 

SECTION   XL. 
PRINCETON. 

[State  what  text-books  you  have  read.] 

1.  (a)  Find  G.  CD.  of 

<6x^+\2>x-h  and  Zx^ -\-2x' -\-'2.x -\. 
(b)   Factor  (nx  —  x -j- y  —  ny) ;  also  (9a^b^  ~-y^). 

2.  Find  the  square  root  of 

x^  +  4:xf  +  y-'  +  4:x^y  —  2x''y-'^  -  4:xK 

3.  Solve  the  equations 


(a/  X      I      (J  X      ,      C         '  X  r\ 

be  ca  ab 


4.    Solve  the  equations 


(b)  (;r-l)-^-5(a;-l)"*  +  l  =  0. 

5.  Solve  the  simultaneous  equations 

=  -  ;    y  —  x^=  ^. 

b-3y     2'   ^ 

6.  Solve  the  simultaneous  equations 

}-'^  =  2]    x  —  y  =  4:. 

y  +  5       X 

7.  What  number  must  be  added  to  m  and  to  n,  in  order 

that  the  sums  may  be  in  the  ratio  of^  to  ^? 


COLLEGE  EEQUIREMENTS  IN  ALGEBRA. 


o>«o 


AITSWEES. 

Section  I.    Page  7. 

1.  ^'. 

452 

3.    ^0.               5.       ^2/ 

2.   4,  5,  6. 

4.    18.              6.    67.68,  32.32. 

7.  10,  lOJ. 


Section  II.     Page  8. 

1.  ^x  +  2y.  6.  a  +  l^ahx+2^h''x+l^hx-2>ahy, 

2-  3a.  ^  28a?- 30?/  + 5a 

3.  5(2a;  +  5  +  c).  *                15 

4.  ^)(a  +  2c).  8.  5.                      9.    li 
6.  -19a:-y.  10.  :r' +  11a;' +  24:r- 36. 


Section  III.     Page  9. 

1.  c  —  ^h.  4.  18a;-29y  +  30.       7.  0.       8.  14. 

2.  -3a.  5.  c{a  —  c).  9.  8. 

3.  ah-ac-b''  +  2hx.    6.  x^^  +  ^x^.  10.  2/'(y  -7). 

1 


COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

Section  IV.     Page  10. 

1.  {x  +  y  +  z){x  +  y-  z){x  —  y  +  z){x  —  y-z). 

2.  {x-l){x^  +  x  +  l){x  +  l)(x'-x  +  l), 

3.  {x  +  ^y){x-^y){x  +  2y){x-2y). 

4.  (Zx  +  2y){2x-2>y). 

5.  (.  +  a3/)(.  +  ^} 

6.  {o?  +  b''  +  c'){a  +  c){a-c), 

7.  (2a  +  35-2c)(3a-25). 

8.  {2a  +  ^h  +  c){a-2h-c). 

9.  {^x^  +  2x'-xy. 

10.  {x'J+^xy  +  y%x'~2>xy  +  fy 

11.  {x'  +  x-yj2  +  V){x''-x^2+\), 

12.  (a;  -  l)(:i;  -  2)(:r  -  3). 

13.  (a7"»  +  i)^ 

14.  [{l+y)-x\l-y)]\ 

Section  V.     Page  11. 

1.  (l  +  x-y){l-x  +  y), 

2.  {a  +  h){a'  -  ah  +  h'){a-h). 

(a^  +  ah  +  h'Xa'  +  h')(a'  -  a'b'  +  b'). 

3.  (x-S)(x+l). 

4.  (4a  +  3)(2a-l). 

5.  (l  +  mx)(l  —  mx)(l  +  nx)(l  —  nx). 


ANSWERS. 

6.  (m  —  n){rn?  +  ^^  +  n^)(x  -{-  y). 

7.  ax{x^  —  X—  \){x  —  \)\ 

8.  (2a-h  +  ^c)(?>a-h  +  c). 

9.  (3:r^-5a;+7y. 

10.  {?>a-2h){a  +  2h)(?>a  +  2h){a-2b). 

11.  ( Va  +  V^)( Va  —  V^). 

12.  (a;+l)(a;  +  4)(^+5). 

13.  (:r  -  y)(^""'  +  ^"~V ^y**"'  +  3/~"')- 

14.  [aa;-(Z)  +  l)?/][(a-l):r  +  5y]. 

Section  YI.  Page  12. 

1.  3a;— 1.  5.    2x''  —  x-b. 

2.  5^  +  2Z^  +  l.  6.    1. 

3.  2a:'-3a;-l.  7.    2a'  +  3a-l. 

4.  2a:'-3a;  +  l.  8.    2a'  +  a-l. 

Section  VII.     Page  13. 

1.  x''  +  2x  +  Z.  5.    1. 

2.  2a; -1.  6.    2.'r''  +  a:~6. 

3.  3a; +  2.  7.    (2x^-x  +  -^V2. 

\  4:xy 

4.  x'  —  x-2.  8.    a;' -2a; -3. 


Section  VIII.     Page  14. 

a:'  +  3a;  +  3* 


1.         .^  +  2  3.    2.  5.    J^.       6.    2. 


4         2a:y  l-.V 

2.   0.  *   (^'  +  y7  *      ^ 


COLLEGE  REQUIREMENTS  IN  ALGEBRA. 


Section  IX.     Page  15. 


abx  x'-{-7x^2 

3     (b-y)(b'-hy+y')       g     ^        ^     (a'  +  a'b'  +  h')(a^by 
{b  +  y)(a'-ay+y')         '      '         '  d'  +  b' 

Section  X.     Page  16. 

1.  5.        3.  12.      5.  5.69i         7.  12al 

2.  5.        4.  6.        6.  -^.         8.  383-2-5-,  5i^,  21^^,  or  54tV 

Section  XL     Page  17. 
1.    4.  3.    2.  5.    5.  7. 


a  —  ^  - 

2.    7,1..  4.   ^.         6.    ""^"-"""^.        8.    200. 

''  25  a^  +  J 


25  a'^  +  5 

Section  XII.     Page  18. 


-  b^c  -  bc^ 

1,    x^ 


ab'  —  a^b 


4.    X  = 


a'b  —  ab' 

cfl  -\-bdn  —  bfm  ^ 
bde  +  acf 

del  4-  afvi  —  adn 

y  =z ' — ^ j 

bde  +  acf 

berti  -\-  acn  —  eel 

z= ■ 

bde  +  aef 


2. 

^— Iff; 

y-^2f|. 

3. 

x  =  a  +  b] 

y  =  a-b. 

5. 

7a^  —  n^ 

X  - 

am  —  on 

m^  —  n^ 

bm  —  an 

6. 

3,  2,  1. 

7. 

Java,  c—b]  Mocha,  a  —  c. 

ANSWFBS. 


Section  XIII.     Page  19. 


1. 

a~  0 
^      a  +  b 

4.    X-     ^  ^          ; 
a-}-  c  —  h 

y-      2      ' 

2. 

x  =  8;    y  =  0. 

a-\-h  —  c 

3. 

2 

5.   x  = 

^2'  ^" 

2'         2 

6.    x  = 

=  2;y  = 

=  -3;  2  =  1J. 

7.    A's 

5c. 
32' 

■D  S  =  -— ■ 

32 

Section  XIV 

.     Page  20. 

3. 

x'y(a'-aY-l) 
d^xy-'  +  xY-^) 

7.    a;*. 

ahh^ 

8.    l  +  2y-i-3y- 

■^  +  4y-\ 

4. 
5. 

2 
a'  -  61 

9.    ^^-2 
a;' +  2 

6.    x^^7/''^+x^y  ^—x  -yi-x  V-    1^-    1^,625. 
Section  XV.     Page  21. 

3. 2^;§±^).  6.  ^:::;^\    7.  a^b^. 

a  o^{a^  —  c*) 


4. 


Q^M 


6. 

x+i    • 
5  « 

8. 

:r*  +  :r-2:r* 

9. 

o;^  —  2  ^^3/^. 

0. 

32. 

COLLEGE  REQUIREMENTS  IN  ALGEBRA. 

Section  XVI.     Page  22. 
^     Q^  6.    12[3  +  V6  +  V2  +  V3]. 


\x-  1 


3, 

1st,  3d,  2d. 

4. 

x'-x+l. 

5. 

a' 

2^x{x  +  y)-2x-y 


y 

8.  {^x'  +  x)^J^^, 

9.  6  +  V5. 
h^^^e                                         10.    ±2. 

Section  XVII.     Page  28. 


1.  (a-l)W  +  a-fl.  6.  V^. 

2.  8  +  ^V5  +  7-v^3.  ^  6V3-F8  +  2V=^ 

3.  3d,  1st,  2d.  ^ 

3_3^_V2  ^-  20V6-62. 

*   "^        4         8  '  9.  2V3  +  V2. 

5.    ■V2.  10.  a;  =  16J. 

Section  XVIII.     Page  24. 


1. 

i  or  -  i. 

6. 

4  or  -v/49. 

2. 

a            —b 

7. 

2Z)  +  9a        2b-a 
2        '^      2a    • 

a  +  b        a  +  o 

3. 

2a  +  b        2a~b 
2a  — b    ^  2a  +  b 

8. 
9. 

±1. 

4. 

1,2,-3,-4. 

10. 

x''-2ax  =  b''-2bc+c^-a\ 

5. 

3. 

11. 

24  or  2 ;  34  or  12. 

ANSWERS. 

Section  XIX.     Page  25. 
1-    tO^-A-  6.    81  or  2401. 

7.    1  or  1  ±  V60. 

2aV3 


2. 

a 

-25  or 

4  b' 

a  +  'Zb 

3. 

a 

a  —  b 

or  -  2. 

4. 

1, 

2,  3,  3. 

r; 

1 

-2 

8. 


3 

i±  V-^ 


9.    -1,  2, 

2  -  V3        2  -  V3  2 

10.    {a-b)x'-4:abx  =  a^  +  a'b-ah'-b\  11.  10  gal. 

Section  XX.     Page  26. 

1.  x  =  Z  or  -i|;  ^     ^_^yiO. 
y  =  2  or  f|.  '  10    ' 

2.  a;  =  9  or  -5;  VTO 
y  =  5  or  -  9.  10 

3.  :r  =  ±l;  '    6.    :r=±V-; 
y  =  ±l.  3/  =  ±i 

4.  :r  =  4  or  —  2;  7.    :?:  —  d=  3  or  ±  |  V2. 
y  =  2or—  4.  3/=±2or±^  V2. 

8.   a;  =  rb9;   2/  =  ±4.  9.   a7=  16  ;  y  =  —  7. 

Section  XXI.     Page  27. 

1.  a;  =  4or— 3;  3.    a:==zb4or±l; 
3/  =  1  or  —  I".  y  =  H=  2  or  =F  3. 

2.  :r  =  7  or  3  ;  4.    :r  =  ±  3  ; 
y  =  —  3  or  —  7.  y  =  ±  3. 


COLLEGE  EEQULREMENTS  IN  ALGEBRA. 


5. 

.r  =  ^  or  |-;                               6.    :r  =  4  or  —  2; 

y  z=  ^  or  i.                                    y  ==  -  2  or  4. 

7.    .-±2  or  ±4^^;    3/-^6or^l^^^. 
3         -^                    ~      3 

8.    :r  =  ±3;    y -- ±  2. 

9.    x^^^  or  -35;    y  =  2a  or  2>(b-a\ 
a  —  b 

Section  XXIL     Page  28. 

3. 

/^    ;    ^^^   .            7.    35.             8.         I 

9.    18. 


•  a 

Section  XXIII.     Page  29. 
2.    :2;>1.      3.    16.      4.    5,3.      7.    346J  sq.  ft.      8.    15  mi. 

Section  XXIV.     Page  30. 
1.    i  =  a  +  {n—l)d.  5.    d-=.-2',  Z=-25. 


2. 

3. 
4. 

rZ  —  a 

6.    -|,  -V-,   ¥• 

'-  r-l 
55U. 

n  +  5 

7.    -64.         8.    f 
9.    3,  4,  5 

3 

10.    300. 

Section  XXV 

.     Page  31. 

1. 

s=l{a  +  l). 

5.    a  =  —  5  or  4  ; 
n  =  43  or  52. 

2. 

l  =  aT''-'\ 

6.    2,  8,  32. 

3. 

-i 

7.    -\^-i-.         9.    8 J. 

4. 

4n-l. 

8.    \.               10.    11,385. 

ANSWEBS. 


Section  XXVL     Page  32. 


32       24  9  27 

2"^  8         16  *  '    4a^~^4a^      16  a** 

4.    a^«+10a^«52  +  ^a^«Z/*  +  ^a'^*5^ 

6.  IM^.  6.    -6a-  7.    54. 

8.  1  +  2nx'  +  2n(n—  l)x^  -\ ^^ ^ ■ — ^ — 

9.  a«  +  3a*  +  6a^  +  6a-2  +  3a-'^-a-«  +  7. 

10.  31.  11.    720.  12.    720. 

Section  XXVIL     Page  33. 

^9  7)6  1AA8 

1.  a'-Sa'b'  +  24:b'-^^+~ 

a^  a* 

2 .  a~^  —  \  oT^x  +  3^  or^x^  —  yVV  <^~"^^V. 

3.  a'-12a'^'"  +  54a*-108a'  +  81a^ 

4.  :r^^ - 100 x^y + 4950 x^'y- 4950 .ry«- 100 xf^  +  ^/^^^ 

g^   __189|^,  ^     -  365,625 a-^^^)^l 

4:r^ 

7.  16a^  +  245  +  ^. 

a/ 

^-    ^  +  [1^+         \2         ^+  13  ^• 

9.    e'^  — 4e''  +  6-4e-'==  +  e-^  10.    1188. 

11.  20.  12.    24  couples;  17,280  arrangements. 


10  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 


Section  XXVIII.     Page  34. 
1  14 


6(a;  +  l)      2(^-1)     Z{x-2) 
4.   § } § - § I-        ^ 


4(a;-l)     4(^--l/     4(:i:  +  l)      ^{x  +  Vf 
5.    0.  7.   -.0000000395145.  8.    .9465. 

Section  XXIX.     Page  35. 

1.    l  +  ?>x  +  ^x^+lx\         2.   ^  +  \x  +  ^-x^-^^x\ 
1  1  1 


3. 


4.a\x--a)     ^a'(x  +  a)     2a\x^  +  a^) 


4. 5 ^—+  1 


3(a;-2)      {x-2y     ^i2x-\) 
5.    1.  7.    3.01814. 

Section  XXX.     Page  36. 

1.  24  to  14.  3.    c  —  ?>a  +  2h. 

2.  I2x-b2w-t-2s.        4.    ia'"+^  +  -|-a-|a'"-^-yV 

5.  a  +  2h-2>c. 

6.  (^>-2ya;2-2(^>2-3^)  +  2)a;  +  (^>-l)^ 

7.  a'  +  a  +  i  +  --  9.    3a: -2. 

a      0/ 

8.  {x +  y  —  z)(x —  y -\- z).    10.    a;^  —  (a  +  5)  a;  +  a5. 

Section  XXXI.     Page  37. 

1.    900or  200  vols. ;  $1800  or  $400. 

ah  +  ac:^2a^Fc  3.    72  or  45  prob. 


2. 


5  —  c  8  or  5,  A. 


ANSWERS.  11 


Section  XXXIL     Page  38. 


1     X-    ^^^     or  ~^^^  2*    a;  =  400  or  -60; 

2a  — b        2a+b  y^i-or— 9. 

3.    (3^-2);    (2>x-2){4:x-l){idx  +  b). 

4.    395a^*^^  5.    2:^2  -  a; -1. 


1. 


Section  XXXIII.     Page  39. 
5V3  -  3  +  4V45  -  4Vl5  -  3 V5 


11 

2.    27a;^-18a;*  +  12a:*-8.  4.    2V3-V15. 

^     rz;  +  3^^  +  3  ^rM^  ^        _  -  a  ±  Va'  -  Z?  -f  7 

o* •  o.     X  — — -• 

l  +  2>x^  +  Zx^  +  x  ^-^ 


6     r-^^  +  ^^  +  ^^~^^-    ..-^V^  +  2a- V^-2a 
2  '    ^  2 

7.    4a-'  -  56a-'a:  +  448a-V  -  2688a-^V  +  13,440a-":r\ 

Section  XXXIV.     Page  40. 

1.    G.C.D.  =  1;  L.C.M.-(2:r+3)(3a:-2)(16rr*+36:r^+81). 
4/a+T 


2     ^7^  "T-  ^  3     ^^^^aVJ.    a;  =  4or  — 2; 

^  +  1  '    y  =  -2or4. 


aS 


4.  :?;^-(a-^>)a;  =  a^.  5.    -f^  ;  81,  1,  0,  ^  0,  2. 

6.    3,  5,  7,  or  15,  5,  -  5.  "^"^ 

Section  XXXV.     Page  41. 
1.    (V^  — 7a)(V^+4a);  2.    a'h^. 

(m'  +  mn  +  n'Xm''  -mn  +  n');       3 .    8  + 1 V5  -  4  V30. 
(ah'^c-'-a'h-'^cy.  5.    -15. 


6.  rr  =  zf=  V3  ;    X  =  a/'8  or  V 

7.  .^'  =  ±4;  y  =  zbl.  8.    $3. 


51  2 

"2T"- 


12  COLLEGE  REQUIREMENTS  IN  ALGEBRA. 


Section  XXXVI.     Page  42. 
2.    {1x  +  l){2x+l)(2x      1);  .r  =  dzi  or  -f 
4.    3V3-2V5;  4.  5.    m==2  or  -\o, 

6.  (a)  a;=.3;  {h)x  =  ^'^  ^^"f'  (^)^  =  ±3;  y^±L 

7.  1.  8.    ^-±-^. 

2 

Section  XXXVII.     Page  43. 
2.    .r  -  1,  y  -  0,  ^  =  0 :  ^  =  y|^  or  0  ;  y  =  yi^  or  0. 


9. 


■g-.  10.    A;=:^ — ^-^^ ^-^ ^;5  =  i;n  =  r— 1. 

71—1 


5 


Section  XXXVIII.     Page  44. 
1     ^(^  +  4)  .   a4-2  4.    :2;  =  2  or  —  8  ;  y  =-  3  or  —  2. 

6.    a:=:27orl.    7.   6.r'-13:r+6  =  0.    8.  2,  6,  10  or  11,  6, 1. 
Section  XXXIX.     Page  45. 


1.  ^^-^^^h'^-^-ac,  j^^vi^. 

2a      ■  

2.  0,-2, -2, +2.  3.    3,  - 1,  or  ^-±-^^^ 

4.    51,300.     5.8,12,16.     6.   l-2x  +  x'  +  a?-2x\      7.0. 

Section  XL.     Page  46. 

1.  Zx-\;  in-V){x-y)-  {?,ah  +  y)(Zab-y). 

2.  af+2x^y-y-K  4.    6  ±  8 V^  ; 

3.  x=^+^+i.  i±r-^^¥. 


^n^ 


«  +  5  +  c  V5  =4=  V21/ 

5.    x  =  —  l\  6.    ri;  =  4or— -|;  pn  —  qm 

y  =  —  3.  y— -Oor— f.  '      Y~~P~' 


MATHEMATICS. 


WENTWORTH'S    SERIES. 

rpHE  United  States  Bureau  of  Education,  in  Circular  of  Infor- 
mation, No.  3,  1890,  says,  on  the  subject  of  mathematical 
text-books,  "If  we  were  called  upon  to  name  the  writer  whose 
books  have  met  with  more  wide-spread  circulation  during  the  past 
decennium  that  those  of  any  other  author,  we  should  answer, 
Wentworth,  ...  In  point  of  scientific  rigor,  Wentworth's  books 
are  superior  to  the  popular  works  of  preceding  decades." 

These  two  points  indicate  the  two  leading  characteristics  of 
these  remarkable  books.  They  combine  mathematical  scholarship 
with  all  the  elements  of  school-room  popularity.  The  author's 
aim  was  to  be  helpful.  He  has  made  a  study  of  his  subjects  with 
the  psychology  and  probable  capacity  of  the  students  constantly  in 
mind.  His  books  enable  the  average  student  not  merely  to  learn 
something,  but  to  master  the  study,  and  at  the  same  time  they 
give  the  brighter  ones  a  plenty  to  exercise  their  faculties  upon. 

There  is  no  attempt  to  make  a  parade  of  learning.  On  the  other 
hand,  there  is  no  absolution  from  solid  work.  Everything  is  made 
simple,  practical,  direct,  and  thorough.  Mental  energy  is  econo- 
mized. The  teacher  has  to  use  his  strength  only  in  the  necessary 
work  of  teaching,  and  the  pupil  his  only  in  the  necessary  work  of 
acquiring. 

To  learn  by  doing,  and  to  learn  one  step  thoroughly  before  the 
next  is  attempted,  are  the  chief  elements  of  the  method,  and  yet 
no  books  give  pupil  or  teacher  less  of  the  tread-mill  feeling.  The 
consciousness  of  mastery  constantly  cheers  and  invigorates  the 
student,  while  the  teacher  has  the  satisfaction  of  wielding  an  in- 
strument fitted  to  the  hand. 

All  will  recognize  in  these  characteristics  the  marks  of  ideal 
text-books.  No  one  will  claim  that  any  series  could  be  suited  to 
every  particular  case.  The  merit  of  Wentworth's  is  that  it  suits 
in  all  but  now  and  then  a  peculiar  one.  The  books  are  found 
everywhere  in  the  United  States,  and  here  and  there  all  around 
the  world.  The  testimony  of  use  may  be  summed  up  in  one  word, 
Satisfaction.     What  more  can  one  desire  than  to  be  satisfied? 


66 


MATHEMATICS. 


Introductory  Books, 


Wentworth^s  Primary  Arithmetic. 

Introduction  price,  30  cents;  allowance  in  exchange,  10  cents. 
Wentwortli^s  Grammar  School  Arithmetic. 

Introduction  price,  65  cents;  allowance  in  exchange,  20  cents. 

These  two  books  form  a  complete  course  in  Arithmetic  for^  Common 
Schools,  and  are  believed  to  present  the  best  known  methods  in  the 
most  attractive,  available,  and  practical  form. 

Wentworth  and  Reed^s  First  Steps  in  Number. 

For  prices,  see  List  at  the  beginning  of  this  Catalogue ;  for  full  descrip- 
tion, see  Common  School  Catalogue. 


A  High  School  Arithmetic, 

(Wentworth  &  Hill's  Practical  Arithmetic.) 

By  G.  A.  Wentworth,  Professor  of  Mathematics,  Phillips  Exetei 
Academy,  and  Dr.  Thomas  Hill,  Portland,  Me.  12mo,  Roan  back. 
367  pages.  Mailing  Price,  ^1.10;  Introduction,  $1.00;  Allowance,  30 
cents;  Answers  free,  on  teacher's  order. 

Same.  Abridged  Edition.  For  Grammar  Schools.  288  pages  {includ- 
ing Answers).  Mailing  Price,  $1.00;  for  introduction,  75  cents;  Allow- 
ance, 25  cents. 

nnniS  book  is  intended  for  high  and  normal  schools  and  acade- 
mies.  It  assumes  that  the  pupil  has  some  knowledge  of  the 
simple  processes  of  Arithmetic,  and  aims  to  develop  his  power  over 
practical  questions,  as  well  as  to  increase  his  facility  in  computing. 
Such  problems  are  selected  as  are  calculated  to  interest  the  pupil, 
and  lead  him  to  independent  thought  and  discovery.  The  prob- 
lems cover  a  wide  range  of  subjects,  and  are  particularly  adapted 
to  general  mental  discipline,  to  preparation  for  higher  studies, 
mechanical  work,  business,  or  professional  life. 


J.  M.  Peirce,  Prof,  of  Mathe- 
matics, Harvard  College:  It  is 
clear,  straightforward,  sound,  care- 
ful, and  abundantly  supplied  with 
well-chosen  examples.  I  would  es- 
pecially commend  the  pains  that 
have  been  taken  to  set  forth  intelli- 
gibly the  shortest  and  easiest  methods 
of  work,  such  as  are  actually  used  by 
the  best  computers,  and  the  inclusion 
of  various  principles  and  rules,  often 
neglected  by  mere  text-book  com- 


pilers, which  go  far  to  form  a  really 
scientific  habit  of  thought  in  the 
young  student. 

H.  Carr  Pritchett,  Prof.  ofMathe* 
matics,  Sam  Houston  Normal  Insti- 
tute,  Tex.:  Wentworth  and  Hill's 
Arithmetic  is  exceedingly  practical. 
All  useless  matter  has  been  discarded, 
and  the  fundamental  principles  so 
clearly  set  forth  that  it  is  a  model  of 
simplicity  and  brevity. 


MATHEMATICS. 


67 


Wentworth's  School  Algebra, 

By  G.  a.  Wentworth,  Professor  of  Mathematics  in  Phillips  Exeter 
Academy.  Half  morocco,  v  +  362  pages.  Mailing  Price,  $1.25;  for 
introduction,  ^1.12;  allowance  for  an  old  book  in  exchange,  35  cents. 
Answers  in  pamphlet  form,  free,  on  teachers'  orders. 

T^  this  book  the  author  has  availed  himself  of  his  own  expe- 
rience in  writing  and  teaching  the  elements  of  Algebra,  and  of 
the  experience  of  hundreds  of  others.  Where  improvement  was 
possible,  it  has  been  made. 

The  School  Algebra  is  offered  as  exactly  right  for  the  usual  high 
school  and  academic  courses.  It  gives  a  thorough  and  practical 
treatment  of  the  principles  of  Algebra  up  to  and  including  the 
binomial  theorem,  and  is  strictly  in  line  throughout  with  the 
author's  College  Algebra.  For  college  preparation  it  is  particu- 
larly well  suited. 

The  problems  in  this  book  are  nearly  all  new,  either  original  or 
selected  from  recent  examination  papers,  and  are  graded  with  the 
utmost  care.  They  are  sufficient  in  number  to  illustrate  and  fix 
all  the  principles,  and  interesting  and  varied  enough  to  hold  the 
student's  attention  through  the  book.  The  passage  from  Arith- 
metic to  Algebra  is  made  easy,  and  the  advantages  of  using  letters 
clearly  pointed  out.  The  treatment  of  fractions  has  been  further 
simplified.  Radicals  precede  quadratics.  There  is  at  the  end  a 
carefully  made  collection  of  miscellaneous  examples,  covering 
nearly  every  principle  of  Algebra.  This,  with  the  author's  College 
Algebra,  makes  a  complete  and  consistent  course. 


J.  B.  Colt,  Prof,  of  Mathematics, 
Boston  University :  In  the  hands  of 
intelligent  teachers,  it  should  lead 
the  young  student  to  pursue  Algebra 
without  feeling  that  it  is  character- 
ized by  arbitrary  laws  and  mysteri- 
ous processes. 

J.  J.  Hardy,  Prof,  of  Mathematics, 
Lafayette  College,  Pa. :  Here  is  an  at- 
tempt by  a  good  teacher,  who  is  also 
familiar  with  the  work  of  great  schol- 
ars, to  make  the  advances  worked 
out  by  them  tell  for  the  improve- 


ment of  elementary  teaching.  The 
result  is  a  most  excellent  book.  It 
is  simple,  yet  scientific;  scholarly, 
yet  an  excellent  drill  book. 

Oscar  Schmiedel,  Prof,  of  Mathe- 
matics, Bethany  College,  W.  Va. :  A 
book  for  beginners,  written  by  a 
teacher  whose  methods  are  clear  and 
concise,  who  understands  the  diffi- 
culties encountered  by  his  pupil,  and 
who  knows  how  to  clear  away  these 
difficulties. 


68 


MATHEMATICS. 


David  Eugene  Smith,  Teacher  of 
Mathematics,  State  Normal  and 
Training  School,  Cortland,  N.Y.: 
I  have  examined  it  with  a  good  deal 
of  care.  It  seems  well  adapted  to 
the  needs  of  our  schools,  —  even  bet- 
ter adapted  than  the  author's  former 
work,  which  I  have  used  and  recom- 
mended. The  improvements  to  he 
found  in  this  work  are  such  as  will 
meet  the  approval  of  all  teachers. 

Erastus  Test,  Prin.  of  Purdue  Uni- 
versity, Prep.  Dept. :  After  a  trial  of 
three  months,  I  am  more  than  satis- 
fied with  the  book.  Barring  a  few 
mistakes  unavoidable  in  a  first  edi- 
tion, I  do  not  see  how  it  can  be 
easily  Improved,  and  am  about  ready- 
to  regard  it  as  the  ne  plus  ultra  in 
the  line  of  a  school  Algebra. 

Prank  E.  Thompson,  Pi-in.  of 
High  School,  Neicport,  R.I.:  I  am 
pleased  with  it,  especially  the  intro- 
ductory chapter  and  the  interpreta- 
tion of  negative  answers.  A  thorough 
knowledge  of  the  contents  of  the 
book  will  enable  a  pupil  to  pass  an 
entrance  examination  to  any  college. 

J.  S.  Slocum,  Prin.  of  South  Divis- 
ion High  School,  Chicago :  I  have 
used  it  in  connection  with  the  prep- 
aration of  a  class  for  college,  and 
have  been  pleased  with  its  clear  defi- 
nitions, logical  arrangement,  and 
happy  selection  of  both  examples 
and  problems. 

Richard  T.  Lewis,  Pres.  of  Judson 
College,  N.C.:  It  is  fresh,  strong, 
really  invigorating,  and  requires 
thought  to  master. 

E.  D.  Sherman,  Prin.  of  High 
School,  Baij  City,  Mich. :  We  think  it 
an  excellent  text-book,  or  we  should 
not  at  present  be  using  it. 


0.  S.  Westcott,  Prin.  of  North 
Division  High  School,  Chicago  :  The 
student  who  finishes  it  will  be  splen- 
didly prepared  to  grapple  with  the 
beautiful  discussions  of  higher  Al- 
gebra. 

Newton  B.  Hobart,  Prin.  of  West- 
ern Reserve  Academy,  Hudson,  0. :  I 
know  of  nothing  better,  and  in  some 
respects  I  consider  it  superior  to  any- 
thing of  the  kind  I  have  seen. 

W.  P.  Durfee,  Prof,  of  Mathemat- 
ics, Hobart  College,  N.Y. :  An  admi- 
rable book  for  college  preparation. 
The  arrangement  of  topics  is  good, 
their  presentation  clear  and  logical, 
and  the  illustrative  examples  all  that 
could  be  desired. 

Cooper  D.  Schmitt,  Prof,  of  Math- 
ematics, University  of  Tennessee  : 
For  the  work  intended,  I  do  not  think 
it  can  be  surpassed. 

W.  N.  Hailmann,  Supt.  of  Schools, 
La  Porte,  Ind. :  For  a  high  school 
using  but  one  book  in  Algebra,  we 
consider  it  decidedly  the  best  in  the 
market. 

C.  L.  Sawyer,  Prin.  of  High 
School,  Waukegan,  III. :  It  has  what 
so  many  books  lack,  —  method. 

George  E.  Gay,  Prin.  of  High 
School,  Maldeyi,  Mass.:  It  is  better 
adapted  to  use  in  high  schools  than 
any  other. 

J.  E.  Smith,  Supt.  of  Schools,  San 
Antonio,  Tex. :  I  think  I  can  con- 
scientiously say  that  it  is  the  best 
high  school  Algebra  that  I  have  ex- 
amined. 

F.  E.  Stratton,  Prin.  of  High 
School,  Davetiport,  la. :  The  author 
evidently  has  the  right  conception 
of  what  is  needed  for  a  high  school 
Algebra. 


MATHEMATICS. 


69 


Wentworth's  College  Algebra, 

By  G.  A.  Wentworth,  Professor  of  Mathematics  in  Phillips  Exeter 
Academy.  Half  morocco.  500  pages.  Mailing  Price,  ^1.65;  for  in- 
troduction, 31.50;  allowance  for  an  old  book  in  exchange,  40  cents. 
Ansioers  in  pamphlet  form,  free ^  on  teachers'  orders. 

rPHIS  is  what  its  name  indicates,  a  text-book  for  colleges  and 
scientific  schools.  The  first  part  is  simply  a  concise  review 
of  the  principles  of  Algebra  preceding  quadratics,  with  enough 
examples  to  illustrate  and  enforce  the  principles.  Room  is  thus 
left  for  a  full  discussion  of  the  higher  topics.  The  endeavor  has 
been  to  give  in  matter  and  methods  the  best  training  in  algebraic 
analysis  at  present  attainable.  The  work  covers  a  full  year,  but 
by  omitting  starred  sections  and  problems,  the  instructor  can 
arrange  a  half-year  course.  Many  professors  helped  with  sug- 
gestions to  make  the  College  Algebra  fit  prevailing  requirements. 


William  Beebe,  Assistant  Prof,  of 
Mathematics  and  Astronomy,  Yale 
College :  I  find  it  characterized  by 
the  clearness  and  method  of  all  Pro- 
fessor Wentworth's  books,  and  am 
particularly  struck  with  the  amount 
of  matter  in  the  Algebra. 

Dascom  Greene,  Prof,  of  Mathe- 
matics and  Astronomy,  Rensselaer 
Polytechnic  Institute,  Troy,  N.Y.: 
The  methods  of  treatment  seem  to 
be  generally  judicious,  and  the  style 
attractive,  and  to  the  student  well 
grounded  in  the  elements  it  furnishes 
an  excellent  practical  course  in  the 
higher  branches  of  the  subject. 

E.  S.  Crawley,  Instructor  in 
Mathematics,  University  of  Penn- 
sylvania :  It  is  a  great  improvement 

over for  college  use,  and  in  its 

way  seems  to  me  to  leave  nothing  to 
be  desired. 

T.  C.  Leonard,  S.  J.,  Prof,  of  Math- 
ematics, St.  Ignatius  College,  San 
Francisco :    The  book,  in  my  opin- 


ion, is  a  model  Algebra,  distinguished 
for  its  clearness  of  explanation  and 
the  eminently  practical  nature  of  its 
matter. 

J.  C.  Glashan,  Inspector  of  Public 
Schools,  Ottaioa,  Canada :  I  am 
satisfied  I  can  unqualifiedly  recom- 
mend it. 

Henry  Kay  Warner,  Prof,  of 
Mathematics,  Mt.  Union  College,  0. : 
Both  as  to  subject-matter  and  style, 
I  regard  it  as  the  most  perfect  Alge- 
bra I  have  ever  examined. 

Henry  C.  King,  Prof,  of  Mathe- 
matics, Oherlin  College,  O.:  The 
chapter  on  Determinants  I  think  a 
specially  valuable  feature,  which 
alone  would  tell  strongly  in  favor 
of  the  book  for  college  use. 

E.  P.  Thompson,  Prof,  of  Mathe- 
matics, Geneva  College,  Pa.:  This 
is  such  a  work  as  the  college  student 
ought  to  use. 


70  MATHEMATICS. 

Wentworth's  Higher  Algebra. 

By  G.  a.  Wentworth,  Professor  of  Mathematics  in  Phillips  Exeter 
Academy.  Half  morocco.  528  pages.  Mailing  price,  31.55;  for  intro- 
duction, 31.40;  allowance  for  an  old  book  in  exchange,  40  cents. 
Answers  in  pamphlet  form,  free,  on  teachers'  orders. 

rnHIS  work  is  designed  to  prepare  thoroughly  for  colleges  and 
scientific  schools,  and  to  furnish  in  addition  what  is  needed  for 
the  general  student  in  such  institutions. 

The  preparatory  schools  and  academies  for  which  it  is  particu- 
larly recommended  are  those  of  high  grade,  especially  such  as  give 
the  pupils  a  thorough  preliminary  drill  in  Arithmetic. 

Normal  schools,  seminaries,  and  a  large  percentage  of  the  col- 
leges, particularly  in  the  West  and  South,  will  find  it  specially 
suited  to  their  requirements. 

It  is  substantially  equivalent  to  the  author's  Complete  Algebra, 
but  is  believed  to  be  in  many  respects  better.  The  arrangement 
and  the  treatment  of  the  topics  have  been  revised,  and  fuller 
explanations  have  been  given. 

As  compared  with  the  School  Algebra,  this  is  more  complete, 
inasmuch  as  it  takes  up  the  topics  usually  included  in  higher 
Algebra. 

As  compared  with  the  College  Algebra,  it  gives  a  fuller  treat- 
ment of  the  simple  elements  of  Algebra  and  not  so  elaborate  a 
treatment  of  the  more  advanced  portions. 

In  a  word,  the  Higher  Algebra  provides  in  a  single  hook  a  com- 
plete course  parallel  to  the  course  provided  by  the  School  and 
College  Algebras  together. 

It  is  not  necessary  to  add  that  it  is  characterized  by  the  special 
excellences  of  Professor  Wentworth 's  other  works  on  this  subject.^ 

As  this  book  is  published  while  the  catalogue  is  printing,  it  is 
impossible  to  present  testimonials. 


WENTWORTH'S    ALGEBRAS. 

Full  Course  :  School  Algebra,  followed  by  College  Algebra. 

Condensed  Course  :  Higher  Algebra. 

Special  Courses  :  Shorter  Course;  Complete  Algebra. 


MATHEMATICS. 


71 


Wentworth's  Elements  of  Algebra. 

By  George  A.  Wentworth,  Professor  of  Mathematics,  Phillips  Exeter 
Academy,  and  author  of  Geometry,  Trigonometry ,  etc.  Half  morocco. 
x  +  325  pages.  Mailing  price,  ^1.25 ;  for  introduction,  ^1.12 ;  allowance, 
35  cents.    Answers  hound  separately  in  pamphlet  form. 

rpHIS  book  is  designed  for  high  schools  and  academies,  and  con- 
tains an  ample  amount  for  admission  to  any  college. 
The  single  aim  in  writing  this  volume  has  been  to  make  an 
algebra  which  the  beginner  would  read  with  increasing  interest, 
intelligence,  and  power.  The  fact  has  been  kept  constantly  in 
mind  that,  to  accomplish  this  object,  the  several  parts  must  be 
presented  so  distinctly  that  the  pupil  will  be  led  to  feel  that  he  is 
mastering  the  subject.  The  chapter  on  Choice,  heretofore  included 
in  the  Elements  of  Algebra,  has  been  omitted  in  accordance  with 
the  wishes  of  many  teachers. 

Wentworth' s  Complete  Algebra, 

Includes  the  subjects  usually  taught  in  Colleges.  Half  morocco.  525 
pages.  Mailing  price,  ^1.55;  for  introduction,  $1.40;  allowance,  40 
cents.    Answers  hound  separately  in  pamphlet  form. 

rpHIS  work  consists  of  the  author's  Elementary  Algebra,  with 
about  one  hundred  and  eighty-five  pages  additional.  The 
additions  are  chapters  on  Choice,  Chance,  Interest  Formulas,  Con- 
tinued Fractions,  Theory  of  Limits,  Indeterminate  Coefficients,  the 
Exponential  Theorem,  the  Differential  Method,  the  Theory  of 
Numbers,  Imaginary  Numbers,  Loci  of  Equations,  Equations  in 
General,  Higher  Numerical  Equations. 


T.  H.  Safford,  Prof,  of  Mathe- 
matics, Williams  College  :  I  am 
using  Wentworth's  Complete  Alge- 
bra to  my  entire  satisfaction. 

A.  S.  Hardy,  Prof,  of  Mathematics, 
Dartmouth  College,  in  New  York 
*^  Critic  " ;  It  is  no  easy  matter  to 
attain  the  true  mean  between  a  dif- 
fuseness  which  enervates  and  a 
brevity  which  discourages.  It  is  not 
too  much  to  say  that  the  author  of 


the  Complete  Algebra  has  been  emi- 
nently successful  in  this  respect,  and 
it  is  safe  to  predict  for  this  work  the 
favorable  reception  which  has  been 
already  awarded  to  his  Geometry. 

Joseph  Hall,  Prin.  of  High  School , 
Hartford,  Conn. :  This  is  the  only 
Algebra  that  we  use  in  preparing 
students  to  enter  the  academic  or 
scientific  department  of  any  college 
or  university. 


72  MATHEMATICS. 

Wentworth's  Shorter  Course  in  Algebra. 

By  George  A.  Wentworth.  Half  morocco.  258  pages.  By  mail,  Sl.lO ; 
for  introduction,  ^1.00;  allowance,  35  cents.  Ansivers  in  pamphlet 
form,  /ree,  on  teachers'  orders. 

rpHIS  book  was  prepared  and  is  recommended  only  for  a  special 
use.  It  was  based  upon  the  author's  Elements  of  Algebra, 
and  contains  all  the  essential  subjects  treated  in  that  book,  but 
with  fewer  examples,  so  as  to  make  a  one  year's  course.  Care  has 
been  taken  to  exclude  all  exacting  problems,  and  yet  the  problems 
are  not  so  easy  as  to  deprive  the  student  of  the  satisfaction  of 
mastery. 

It  is  a  significant  recommendation  of  the  Shorter  Course  in 
Algebra  that  it  was  recently  adopted  for  exclusive  use  in  all  the 
schools  of  the  State  of  Washington. 

C.  K.  Wells,  Svpt.  of  Schools, 
Marietta,  0.:  It  has  given  good  sat- 
isfaction. It  is  full  enough  for  a 
high  school  course.  I  think  it  will 
hold  its  ground  in  our  high  school 
for  a  long  time. 

0.  T.  Snow,  Prin.  of  High  School^ 
Batavia,  III. :  It  is  the  best  treatise 
on  the  subject  I  have  ever  seen.  I 
shall  recommend  its  adoption. 


Alfred  S.  Rowe,  late  Prin.  of  High 
School,  Worcester,  Mass. :  I  am  sat- 
isfied that  it  will  meet  a  popular 
demand.  It  is  not  so  difficult  as  to 
discourage,  nor  so  easy  as  to  give  a 
misimpression  of  the  character  of 
the  study. 

B.T.McCord,  Prof,  of  Mathematics, 
Lincoln  University,  III. :  I  think  it  is 
admirably  adapted  for  a  short  course. 

Algebraic  Analysis. 

By  G.  a.  Wentworth,  A.M.,  Professor  of  Mathematics  in  Phillips  Exe- 
ter Academy;  J.  A.  McLellan,  LL.D.,  Inspector  of  Normal  Schools, 
and  Conductor  of  Teachers'  Institutes  for  Ontario,  Canada ;  and  J.  C. 
Glashan,  Inspector  of  Public  Schools,  Ottawa,  Canada.  Part  I.  con- 
cluding with  Determinants.  12mo.  Half  leather,  x  +  418  pages.  By 
mail,  f  1.60 ;  to  teachers  and  for  introduction,  $1.50. 

nPHIS  w^ork,  which  has  been  previously  announced  as  Wentworth 
§•  McLellan' s  University  Algebra,  is  intended  to  supply  students 
of  Mathematics  with  a  well-filled  storehouse  of  solved  examples 
and  unsolved  exercises  in  the  application  of  the  fundamental  theo- 
rems and  processes  of  pure  Algebra,  and  to  exhibit  to  them  the 
highest  and  most  important  results  of  modern  algebraic  analysis. 

J.  M.  Taylor,  Prof,  of  Mathemat-   examples,  and  clearly  presents  the 
ics,   Colgate    University,    N.Y. :   It   best  methods  of  their  solution, 
contains  an  admirable  collection  of 


MATHEMATICS.  73 

Wentworth's  New  Plane  Geometry , 

By  George  A.  Wentworth,  Teacher  of  Mathematics,  Phillips  Exeter 
Academy,  N.H.  12mo.  x  +  242  pages.  Mailing  Price,  85  cents ;  Intro- 
duction, 75  cents ;  Allowance  for  old  book,  25  cents. 

Wentworth' s  New  Plane  and  Solid  Geometry. 

By  George  A.  Wentworth,  Phillips  Academy,  Exeter,  N.H.  12mo. 
Half  morocco,  xi  +  386  pages.  Mailing  Price,  $1.40;  Introduction,  $1.25 ; 
Allowance  for  old  book,  40  cents.  The  book  now  includes  a  treatise  on 
Conic  Sections  (Book  IX.). 

A  LL  the  distinguishing  characteristics  of  the  first  edition  have 
been  retained.  The  subject  is  treated  as  a  branch  of  practical 
logic,  the  object  of  which  is  to  detect  and  state  with  precision  the 
successive  steps  from  premise  to  conclusion. 

In  each  proposition  a  concise  statement  of  what  is  given  is 
printed  in  one  kind  of  type,  of  what  is  required  in  another,  and 
the  demonstration  in  still  another.  The  reason  for  each  step  is 
indicated  in  small  type  between  that  step  and  the  one  following ; 
and  the  author  thus  avoids  the  necessity  of  interrupting  the  process 
of  demonstration  to  cite  a  previous  proposition.  The  number  of 
the  section  on  which  the  reason  depends,  is,  however,  placed  at  the 
side  of  the  page ;  and  the  pupil  should  be  prepared,  when  called 
upon,  to  give  the  proof  of  each  reason.  Each  distinct  assertion  in 
the  demonstrations  and  each  particular  direction  in  the  construc- 
tion of  the  figures  begins  a  new  line,  and  in  no  case  is  it  necessaiy 
to  turn  the  page  in  reading  a  demonstration. 

In  the  new  edition  will  be  found  a  few  changes  in  the  order  of 
the  subject-matter.  Some  of  the  demonstrations  have  been  given 
in  a  more  concise  and  simple  form.  The  diagrams,  with  which 
especial  care  was  taken  originally,  have  been  re-engraved  and  mate- 
rially improved.  The  shading,  which  has  been  added  to  many  of 
the  figures,  has  proved  a  great  help  to  the  constructive  imagination 
of  pupils.  The  theory  of  limits  —  the  value  of  which  the  author 
emphasizes  —  has  been  presented  in  the  simplest  possible  way,  and 
its  application  made  easy  of  comprehension. 

But  the  great  feature  of  this  edition  is  the  introduction  of  nearly 
seven  hundred  original  exercises,  consisting  of  theorems,  problems 
of  construction,  and  problems  of  computation,  carefully  graded  and 
adapted  to  beginners  in  Geometry. 


74 


MATHEMATICS. 


George  W.  Sawin,  late  Instr.  in 
Mathematics  in  Harvard  College : 
The  old  edition  of  Wentworth's 
Geometry,  at  least  the  part  devoted 
to  Plane  Geometry,  I  always  re- 
garded as,  on  the  whole,  the  best 
text-book  of  its  kind  in  English. 
The  new  edition  is,  in  my  opinion,  a 
far  better  book  than  the  old.  The 
books  on  Solid  Geometry  have  been 
raised  to  the  excellence  of  the  first 
five  books.  Even  with  a  teacher  of 
very  moderate  abilities,  this  text- 
book ought  to  render  the  study  of 
Geometry  attractive,  and,  I  may 
add,  fascinating,  to  the  student  of 
average  talents.  It  may  interest 
you  to  learn  that  I  found  last  year 
that  out  of  a  class  of  one  hundred 
and  ten  in  Solid  Geometry ^  one  hun- 
dred and  one  had  been  prepared  in 
Wentworth's  Plane  Geometry, 
(Nov.  13, 1888.) 

W.  A.  Moody,  Prof,  of  Mathemat- 
ics, Boiodoin  College :  I  have  exam- 
ined it  with  considerable  care,  and 
consider  it  an  improvement  on  the 
old  edition,  noticeably  in  the  better 
figures  and  general  finer  appearance 
of  the  book  and  in  the  numerous 
"  original "  exercises. 

J.  R.  French,  Prof,  of  Mathemat- 
ics, Syracuse  University :  It  seems 
to  be  a  great  improvement  upon  the 
former  edition.  We  shall  doubtless 
continue  to  use  it, 

W.  C.  Bartol,  Prof  of  Mathe- 
matics, Bucknell  University :  I  be- 
lieve it  is  a  decided  improvement  on 
the  old  edition,  and  shall  use  it  in 
my  classes. 

Lyman  Hall,  Prof,  of  Mathematics, 
Georgia  School  of  Technology :  I  find 
the  new  edition  of  Wentworth's  Ge- 
ometry admirably  adapted  to  our 
needs  here.    I  think  the  author  has 


accomplished  the  great  end  of  giving 
clear  and  concise  proofs  in  the  mini* 
mum  of  space,  and  has  materially 
improved,  without  rendering  too  diffi- 
cult, the  original  exercises  and  nu- 
merical examples. 

E.  H.  Stanley,  Instructor  of  Math^ 
ematics,  Oherlin  College:  Quite  a 
careful  examination  of  the  Geometry 
has  led  me  to  regard  it  one  of  the 
very  best  of  its  kind. 

W.  Hoover,  Prof  of  MathemMics, 
Ohio  University :  I  have  introduced 
into  my  classes  the  new  edition  of 
Wentworth's  Geometry,  and  find  it 
very  much  of  an  improvement  upon 
the  old.  ...  I  never  had  such  good 
results  in  my  geometry  classes. 

E.  Miller,  Prof  of  Mathematics, 
University  of  Kansas :  The  book  is 
a  very  superior  one,  and  grows  in  ou" 
estimation  because  it  is  exactly  suited 
to  our  needs.  It  is  a  book  that  com- 
pels a  student  to  think  and  invent 
new  methods  and  demonstrations. 

L.  S.  Hulburt,  Prof,  of  Mathe- 
matics, University  of  Dakota:  The 
revised  edition  is  the  best  text-book 
on  the  subject  of  geometry  that  I 
have  ever  used. 

J.  M.  Taylor,  Prof,  of  Mathematics, 
University  of  Washington  Territory: 
The  test  book  on  geometry  has  been 
made  better  by  the  revision. 

A.  S.  Roe,  lately  Prin.  of  High 
School,  Wo7'cester,  Mass. :  I  wish  to 
express  my  unqualified  approval  of  it. 

S.  Weimer,  Teacher  of  Mathemat- 
ics.  High  School,  Cleveland,  Ohio: 
The  new  book  is  a  decided  improve- 
ment over  the  old,  although  his  first 
book,  which  we  are  now  using,  is 
superior  to  any  text-book  for  class 
work  that  I  have  any  knowledge  ot 


MATHEMATICS.  75 

Wentworth's  Trigonometries. 

''TIHE  aim  has  been  to  furnish  just  so  much  ef  Trigonometry  as 
is  actually  taught  in  our  best  schools  and  colleges.  The  prin» 
ciples  have  been  unfolded  with  the  utmost  brevity  consistent  with 
simplicity  and  clearness,  and  interesting  problems  have  been 
selected  with  a  view  to  awaken  a  real  love  for  the  study.  Much 
time  and  labor  have  been  spent  in  devising  the  simplest  proofs  foi 
the  propositions,  and  in  exhibiting  the  best  methods  of  arranging 
the  logarithmic  work.     Answers  are  included. 

Plane  and  Solid  Geometry,  and  Plane  Trigonometry. 

12mo.    Half  morocco.    490  pages.    Mailing  Price,  $1.55;  Introduction, 
$1.40;  Allowance  for  old  book,  40  cents. 

Plane  Trigonometry, 

12mo.    Paper.  80  pages.    Mailing  Price,  35  cents;  Introduction,  30 cents. 

Plane  Trigonometry  Formulas, 

Two  charts  (30  x  40  inches  each)  for  hanging  on  the  walls  of  the  class- 
room.   Introduction  Price,  $1.00  per  set. 

Plane  Trigonometry  and  Logarithms. 

8vo.     Cloth.    160  pages.    Mailing  price,  85  cents ;  introduction,  80  cents. 

Plane  and  Spherical  Trigonometry, 

12mo.     Half  morocco,    iv  +  151  pages.     Mailing  Price,  80  cents;  foi 
introduction,  75  cents ;  allowance,  20  cents. 

Plane  and  Spherical  Trigonometry,  with  Tables, 

8vo.    Half  morocco,    vi  +  259  pages.    Mailing  Price,  $1.25 ;  for  intro- 
duction, $1.12.    Allowance  for  old  book,  35  cents. 

WentworWs  Plane  and  Spherical  Trigonometry 

and  Surveying,  with  Tables. 

Svo.    Half  morocco.    307  pages.    Mailing  Price,  $1.40;  Introduction, 
$1.25 ;  Allowance  for  old  book,  40  cents. 

Surveying, 

Svo.    80  pages.    Paper.    Mailing  Price,  35  cents  ^  for  introduction,  30 
cents. 

Went  worth's  Plane  and  Spherical  Trigonometry, 

Surveying,  and  Navigation, 

12mo.    Half  morocco.    359  pages.    Mailing  Price,  $1.25;  Introduction. 
$1.12;  Allowance  for  old  book,  35  cents. 


76 


MATHEMATICS. 


npHE  object  of  the  work  on  Surveying  and  Navigation  is  to  pre- 
sent these  subjects  in  a  clear  and  intelligible  way,  according 
to  the  best  methods  in  actual  use ;  and  also  to  present  them  in  so 
small  a  compass,  that  students  in  general  may  find  the  time  to 
acquire  a  competent  knowledge  of  these  very  interesting  and 
important  studies.     Answers  are  included. 


A.  H.  Pierce,  Instructor  in  Mathe- 
matics, Amherst  College  :  I  consider 
Wentworth's  Trigonometry  a  perfect 
book  for  the  class-room.  All  unnec- 
essary matter  is  omitted,  and  the  ar- 
rangement of  the  work  is  such  as  to 


help  a  student  to  a  clear  outline  of 
the  whole  subject,  .  .  .  and  the  plen- 
tiful supply  of  exercises  and  practical 
problems  relieves  the  teacher  of  the 
necessity  of  constantly  consulting 
other  text-books. 


Wentworth  &  Hiirs  Five-Place  Logarithmic  and 

Trigonometric  Tables, 

By  G.  A.  Wentworth,  A.M.,  and  G.  A.  Hill,  A.M. 

Seven  Tables  (for  Trigonometry  and  Surveying):  Cloth.   8vo.    79  pages. 
Mailing  Price,  65  cents;  Introduction,  50  cents. 

Complete  (for  Trigonometry,  Surveying,  and  Navigation) :  Half  mo 
rocco.    8vo.    158  pages.    Mailing  Price,  $1.10;  Introduction,  $1.00. 

nnHESE  Tables  have  been  prepared  mainly  from  Gauss's  Tables, 
and  are  designed  for  the  use  of  schools  and  colleges.  They 
are  preceded  by  an  introduction,  in  which  the  nature  and  use  of 
logarithms  are  explained,  and  all  necessary  instruction  given  for 
using  the  tables.  They  are  printed  in  large  type  with  very  open 
spacing.  Compactness,  simple  arrangement,  and  figures  large 
enough  not  to  strain  the  eyes,  are  among  the  points  in  their  favor. 


Wentwortj]  &  Hill's  Exercises  in  Arit/imetie. 

I.  Exercise  Manual.  12mo.  Boards:  vi  +  282  pages.  Mailing  Price, 
65  cents;  for  introduction,  50  cents.  II.  Examination  Manual.  12mo. 
Boards.  148  pages.  Mailing  Price,  40  cents ;  Introduction  Price,  35  cents, 
Both  in  one  volume,  80  cents.    Ansivers  to  both  parts  together ,  10  cents. 

rpHE  first  part  (Exercise  Manual)  contains  3869  examples  and 
problems  for  daily  practice,  classified   and  arranged  in  the 
common  order;  and  the  second  part  (Examination  Manual)  con- 
tains  300  examination-papers,  progressive  in  character. 


MATHEMATICS. 


77 


Wentworth  &  Hill's  Exercises  in  Algebra. 

I.  Exercise  Manual.  12mo.  Boards.  232  pages.  Mailing  Price,  40 
cents;  Introduction  Price,  35  cents.  II.  Examination  Manual.  12mo. 
Boards.  159  pages.  Mailing  Price,  40  cents;  Introduction  Price,  35 
cents.  Both  in  one  volume,  70  cents.  Answers  to  both  parts  together, 
25  cents. 

rpHE  first  part  (Exercise  Manual)  contains  about  4500  problems 
classified  and  arranged  according  to  the  usual  order  of  text- 
books in  Algebra;  and  the  second  part  (Examination  Manual) 
contains  nearly  300  examination-papers,  progressive  in  charao 
ter,  and  well  adapted  to  cultivate  skill  and  rapidity  in  solving 
problems. 


British  Mail:  All  engaged  in  the 
practical  woik  of  education  will 
appreciate  these  Manuals,  as  they 
are  calculated  to  save  the  master 


much  precious  time  and  labor,  and 
to  give  his  students  the  benefit  oi 
progressive  and  carefully  thought* 
out  exercises. 


Wentworth  &  Hill's  Exercises  in  Geometry. 

12mo.    Cloth.    255  pages.    Mailing  Price,  80  cents ;  Introduction  Price, 
70  cents. 

npHE  exercises  consist  of  a  great  number  of  easy  problems  for 
beginners,  and  enough  harder  ones  for  more  advanced  pupils. 
The  problems  of  each  section  are  carefully  graded,  and  some  of  the 
more  difficult  sections  can  be  omitted  without  destroying  the  unity 
of  the  work.  The  book  can  be  used  in  connection  with  any  text- 
book on  Geometry  as  soon  as  the  geometrical  processes  of  reason- 
ing are  well  understood. 


select  propositions  from  it  to  supple- 
ment every  stage  of  our  work. 


Amelia  W.  Platter,  High  School, 
Indianapolis,  Ind. :  I  find  the  sub- 
ject so  carefully  graded,  that  I  can 

Analytic  Geometry. 

By  G.  A.  Wentworth.  Revised  edition,  12mo.  Half  morocco,  xii  + 
301  pages.  Mailing  Price,  $1.35;  for  introduction,  $1.25;  allowance  in 
exchange,  30  cents. 

nnHE  aim  of  this  work  is  to  present  the  elementary  parts  of  the 
subject  in  the  best  form  for  class-room  use. 
The  connection  between  a  locus  and  its  equation  is  made  per 
fectly  clear  in  the  opening  chapter. 


78 


MATHEMATICS. 


The  exercises  are  well  graded,  and  designed  to  secure  the  best 
mental  training.  By  adding  a  supplement  to  each  chapter,  the 
author  has  made  provision  for  a  shorter  or  more  extended  course, 
as  the  time  given  to  the  subject  will  permit. 


Dascom  Greene,  Prof,  of  Mathe- 
matics and  Astronomy,  Rensselaer 
Polytechnic  Institute,  Troy,  N.Y.: 
It  appears  to  be  admirably  adapted 
to  the  use  of  beginners,  and  is  espe- 
cially rich  in  examples  for  practical 
application  of  the  principles  of  each 
chapter.  The  full  and  clear  explana- 
tion of  first  principles  given  in  the 
opening  chapter  is  a  new  and  highly 
commendable  feature  of  the  work. 


E.  Miller,  Prof  of  Mathematics, 
University  of  Kansas,  Lawrence : 
As  a  book  for  beginners,  it  is  admi- 
rable in  all  its  arrangements  and 
features.  The  great  number  of 
problems  scattered  through  it  will 
largely  relieve  it  of  that  abstract 
analysis  which  is  so  often  a  terror 
to  students.  The  book  is,  like  the 
other  works  of  Professor  Went  worth, 
a  good  thing. 


Elementarij  Mathematical  Tables. 


By  Alexander  Macfarlane,  D.Sc,  LL.D.,  Professor  of  Physics  in  the 
University  of  Texas.  8vo.  Cloth,  iv  + 10(5  pages.  Mailing  price,  85 
cents ;  for  introduction,  75  cents. 

rPHIS  collection  of  tables  contains  logarithms,  antilogarithms, 
addition  and  subtraction  logarithms,  logarithmic  sines  and 
cosines,  tangents  and  cotangents,  natural  sines  and  cosines,  tan- 
gents and  cotangents,  secants  and  cosecants ;  arcs,  reciprocals, 
squares,  cubes,  square  roots,  cube  roots,  circumferences,  circular 
areas,  spherical  contents,  powers,  constants,  hyperbolic  logarithms, 
exponentials,  divisors,  least  divisors,  interest  tables,  and  a  large 
number  of  auxiliary  tables.  The  tables  are  mostly  four-place; 
they  have  a  uniform  decimal  arrangement,  and  have  been  made 
every  way  convenient  and  adequate. 

elegant  arrangement.  The  book  is 
excellently  adapted  for  use  in  every 
form  of  computation  where  the  more 
elaborate  tables  are  not  demanded. 


J.  B.  Colt,  Prof,  of  Math.,  Boston 
Univ.,  Boston,  Mass. :  They  are 
surely  worthy  of  very  high  commen- 
dation. I  am  impressed  with  the 
amount  of  valuable  material  and  the 


{Jan.  15,  1890.) 


The  Algebra  of  Logic. 


With  examples.  By  Alexander  Macfarlane,  Professor  of  Physics 
in  the  University  of  Texas.  12mo.  Cloth.  Illustrated  with  diagrams. 
x  +  155  pages.  By  mail,  ^1.35;  for  introduction,  31.25.  The  principles 
of  the  Algebra  of  Quality  investigated  and  compared  with  the  principles 
of  the  Algebra  of  Quantity. 
Westminster  Review,  London :  It  will  find  eager  and  attentive  readers. 


MATHEMATICS. 


79 


A  Treatise  on  Plane  Surveying, 

By  Daniel  Carhart,  C.E.,  Professor  of  Civil  Engineering  in  the  West> 
ern  University  of  Pennsylvania,  Allegheny.  Illustrated.  8vo.  Half 
leather,   xvii  +  498  pages.    Mailing  Price,  $2.00;  for  introduction,  $1.80. 

rriHlS  work  covers  the  whole  ground  of  Plane  Surveying.  It 
illustrates  and  describes  the  instruments  employed,  their  ad- 
justments and  uses ;  it  exemplifies  the  best  methods  of  solving  the 
ordinary  problems  occurring  in  practice,  and  furnishes  solutions 
for  many  special  cases  which  not  infrequently  present  themselves. 
It  is  the  result  of  twenty  years*  experience  in  the  field  and  technical 
schools,  and  the  aim  has  been  to  make  it  extremely  practical,  having 
in  mind  always  that  to  become  a  reliable  surveyor  the  student  needs 
frequently  to  manipulate  the  various  surveying  instruments  in  the 
field,  to  solve  many  examples  in  the  class-room,  and  to  exercise 
good  judgment  in  all  these  operations.  Not  only,  therefore,  are 
the  different  methods  of  surveying  treated,  and  directions  for  using 
the  instruments  given,  but  these  are  supplemented  by  various  field 
exercises  to  be  performed,  by  numerous  examples  to  be  wrought, 
and  by  many  queries  to  be  answered. 

The  judicial  functions  of  surveyors,  as  given  by  Chief  Justice 
Cooley,  are  set  forth  in  an  appendix. 

As  a  practical  and  complete  treatise,  Carhart's  Surveying  has 
received  a  cordial  welcome. 


"W.  A.  Moody,  Prof,  of  Mathemat- 
ics, Bowdoin  College :  I  consider  the 
book  exceptionally  fine  in  execution, 
subject-matter,  and  arrangement. 

D.  W.  Hering,  formerly  Prof,  of 
Math.,  Univ.  of  City  of  New  York: 
The  Surveying  is,  I  think,  superior 
as  a  text-book  to  any  book  on  the 
subject  with  which  I  am  acquainted. 
It  is  compendious  without  being  too 
voluminous,  and  the  skilful  treat- 
ment of  the  subject  accords  perfectly 
with  the  methods  of  the  author,  both 
as  a  teacher  and  a  practical  engi- 
neer. 

Oren  Root,  Prof,  of  Mathematics, 
Hamilton  College :  I  have  looked  it 


through  with  great  interest.  The 
mechanical  execution  is,  in  the  first 
place,  elegant;  the  arrangement  is 
admirable.  .  .  .  The  work  seems  ad- 
mirably adapted  to  student  use  and 
the  class-room. 

Wm.  Hoover,  Prof,  of  Mathemat- 
ics, Ohio  University:  It  is  indeed  a 
superior  work,  and  merits  the  widest 
adoption. 

Colman  Bancroft,  Prof,  of  Mathe- 
matics, Hiram  College :  I  find  in  it 
several  important  matters  not  con- 
tained in  other  text-books  with  which 
I  am  acquainted,  —  matter  which  I 
have  felt  obliged  t«  give  my  classes 
by  lectures. 


80 


MATHEMATICS. 


Hiirs  Geometry  for  Beginners. 

By  G.  A.  Hill,  A.M.  12mo.  Cloth.  320  pages.  Mailing  price,  ^I.IO; 
for  introduction,  ^1.00;  allowance,  30  cents.  Ansivers,  in  pamphlet 
fovrriy  can  he  had  by  teachers. 

n^HIS  book  presents  the  subject  in  the  natural  method  as  distin- 
guished from  the  formal  method  of  Euclid,  Legendre,  and  the 
common  text-books.     The  central  purpose  is  intellectual  training, 
or,  teaching  hy  practice  how  to  think  correctly  and  continuously. 


W.  E.  Byerly,  Prof,  of  Mathe- 
matics, Harvard  University  :  I  do  not 
see  how  the  part  devoted  to  plane 
geometry  could  be  improved. 

John  Trowbridge,  Prof,  of  Mathe- 
matics, Harvard  University:  If  I  had 


been  taught  geometry  in  the  manner 
set  forth  by  Mr.  Hill,  I  should  have 
been  saved  at  least  six  years  of  blun- 
dering effort  in  endeavoring  to  grasp 
the  subject  of  geometry  from  set 
propositions. 


Hill's  Lessons  in  Geometry. 


For  the  Use  of  Beginners.  By  G.  A.  Hill,  A.M.,  author  of  the  Geornetry 
for  Beginners.  12mo.  Cloth.  190  pages.  Mailing  price,  75  cents ;  for 
introduction,  70  cents ;  allowance  for  an  old  book  in  exchange,  24  cents. 
Answers,  in  pamphlet  for  in,  can  he  had  hy  teachers. 

rpmS  is  a  course  similar  to  that  given  in  the  Geometry  for  Begin- 
ners, but  it  is  shorter  and  easier,  and  does  not  require  a  knowl- 
edge of  the  metric  system.  Like  the  Geometry  for  Beginners  this  is 
especially  commended  to  those  who  cannot  pursue  the  study  far 
but  desire  the  discipline  of  geometry. 


C.  C.  Rounds,  Prin.  State  Normal 
School,  Plymouth,  N.H.:  For  giving 
to  students  who  have  never  studied 
geometry  a  real  and  living  knowl- 
edge of  the  subject  and  a  command 
of  its  more  important  applications, 
I  know  of  no  book  equal  to  this. 


Corwin  F.  Palmer,  Supt.  of  Schools, 
Dresden,  Ohio:  It  is  a  delightful 
litde  work,  and  full  of  inspiration. 
A  teacher  who  gets  the  spirit  of  that 
book  into  him  cannot  fail  to  teach 
well.  In  the  hands  of  the  pupil  I 
know  of  nothing  that  approaches  it. 


Hill's  Drawing  Case. 


Prepared  expressly  to  accompany  Hill's  Lessons  in  Geometry,  and  con- 
taining, in  a  neat  wooden  box,  a  seven-inch  rule  with  a  scale  of  milli- 
meters ;  pencil  compasses,  with  pencil  and  rubber ;  a  triangle ;  and  a 
protractor.  Retail  price,  40  cents ;  for  introduction,  30  cents. 
A  specimen  copy  of  the  Lessons  in  Geometry  with  the  Drawing  Case 
will  be  sent,  postpaid,  to  any  teacher  on  receipt  of  $51.00. 


